Are there still 'everyday' phenomena unexplained by Physics? There are two very famous quotes from German Nobel Laureate Albert Abraham Michelson that are remembered mainly for being extremely wrong (especially since he said them just before two major revolutions in physics, quantum mechanics and relativity):


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*The more important fundamental laws and facts of physical science have all been discovered, and these are so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote.

*Many other instances might be cited, but these will suffice to justify the statement that “our future discoveries must be looked for in the sixth place of decimals.” 
It is however somewhat understandable that Michelson thought physics was almost 'completed'. Almost every physics-related phenomenon that a human could encounter in day-to-day life had been explained, including gravity, motion, and electromagnetism.
With the advent of quantum mechanics even more physical phenomena have been explained. It has gotten to the point nowadays that to a layman, it might seem that physics is indeed 'complete'. As far as I know, the exceptions to this rule lie very deep within the realm of theoretical physics, e.g. in topics such as quantum gravity, dark matter, or dark energy. These are things that the average person doesn't know a thing about. Furthermore, and in contrast to classical mechanics, he really doesn't need to know about them, since they mostly involve worlds that are very small, very big, very far away, or very hypothetical. 
This leads me to my question:
Are there any 'everyday' phenomena that remain unexplained by physics? 
To clarify, by everyday I mean regarding 'stuff' that the average person knows something about, and might perhaps encounter in everyday life. For example, an unresolved issue in mechanics might qualify even if it is not a common effect.
 A: Ball lightning.
Ball lightning is exceedingly rare of course, but unlike, say, dark matter, it is possible to encounter ball lightning in the course of everyday life.
My understanding is that in the case of ball lightning it's not even known whether our current physics laws can account for it.
A: High-critical temperature superconductivity is an unexplained physical phenomenon that is close to becoming 'everyday' life. Superconductivity in general has been in use for medical purposes, such as magnetoencephalography, for several years. Recently there have been experimental realizations of magnetoencephalography with high-critical temperature superconducting devices (preprint of published paper).
High-critical temperature superconductivity is also in the near future of public transport using magnetic levitation. Even though the first such trains are based on conventional superconductivity, which is well understood, small-scale prototypes employing unconventional superconductors already exist.
At this point, a note should be made about quantum mechanics in general. Superconductivity is a macroscopically observable quantum phenomenon. Its quantum mechanical description allows us to predict material properties and subsequently come up with technological applications. This, however, does not mean that quantum mechanics itself (or any physical theory, for that matter) is established beyond all doubt as the correct theory. It is just the most consistent microscopic theory we have today. The extent to which we understand quantum-mechanical phenomena is therefore bounded by the inherent suppositions of quantum mechanics (e.g. the Pauli principle, as discussed in Terry Bollinger's answer).
A: A Sizable Mystery
Here's a mystery that remains poorly understood, though there have been many attempts to explain it:

Why does volume -- the ability of matter to fill up space exclusively -- depend on how particles rotate?

By volume I mean for example the fact that you can pound on a desk with your fist, and your fist stops at the desk. The matter in your desk and your fist exclude each other from occupying the same space. Without volume, the universe would be a very boring place. That's because instead of planets, suns, and nebulae we would have black holes, black holes, and black holes. Furthermore, the same features that enable volume also enable all the incredible richness and variety of combination called chemistry. So, without the physics of volume, we would not be here to talk about the topic in the first place.
For Every Volume, Turn, Turn, Turn
Yet the existence of volume depends rather remarkably on the way some particles rotate.  It is that simple connection to rotation that remains mysterious and still smells of something important being overlooked, of some insight that if finally found would make everyone go "Ah! So that's what's really going on there!" But that simple insight remains missing, even though folks such as Nobel Laureate Richard Feynman worked on the problem off and on for decades, without any notable success.
I should emphasize first that how volume works is very well understood.
An Exclusive Club
It's created by something called the Pauli exclusion principle, which behaves like an extremely powerful repulsive force that only comes into effect when identical particles of a certain type, called fermions, are pressed closely together. Fermions are what we usually think of as matter, and they have an "address space" with three parts: location, momentum, and spin orientation (think axis of a spinning globe). As long as all particles remain unique in at least one part of this address space, the fermions are happy, which is to say they stay fairly low in energy. All of the geometry and bonding mechanisms of chemistry arise directly from the rather complicated interplay of a nucleus that attracts a set of electrons, and of all of those electrons insisting on having their own unique three-component addresses.
But that is the known part. The hard-to-explain-well part is why Pauli exclusion is experimentally tied to a very specific type of particle rotation.
As with many quantities in quantum mechanics, the rotation of a very small object begins to lock into discrete values that are based on their angular momentum. Realizing that this quantization would have to occur for angular momentum, physicists defined the smallest unit of angular momentum as spin 1. No one really thought much about it at first, since spin just seemed like yet another "feature" that needed to be tracked when talking about atoms and particles.
A Tale of Two Particle Types
This assumption turned out to be spectacularly wrong. It was subsequently realized from experimental data that the entire universe seems to break down into two major classes of particles, and that these two classes are based entirely on how they spin. The first group is the fermions I've already talked about; they are the ones that have Pauli Exclusion and thus volume.
The second group is called the bosons. The bosons have spins that are simple integer multiples of the smallest obvious unit of quantized spin, spin 1. But these fundamental particles have no volume! They not only don't care at all if they share the same address, there are cases where they prefer to have the same address. That is what a laser is: A lot of spin 1 particles of light that have decided to join together and all occupy the same location, momentum, and spin address at one time. Fundamental bosons are what we usually think of as some form of energy.
But if bosons have rotations that are simple multiples of the smallest possible unit of rotation, spin 1, what kind of rotation can fermions have that is different? Where do they fit in?
The Sound of Half a Rope Spinning
That is the first really weird thing about volume: It is based on particles whose rotations are offset by exactly $\frac{1}{2}$ unit from the boson rotation  values, and thus fit "in between" the integer spin values of the bosons. So for example, fundamental electrons and the more complicated protons and neutrons of matter all have spin $\frac{1}{2}$, and so all occupy space.
If all that sounds odd, it is. The fermion offset of "spin $\frac{1}{2}$" was completely unanticipated to theorists. It was first a source of amusement and then bafflement when experimentation first forced theorist to consider its existence. For a theorist of that time (or now!), trying to interpret the visual meaning of "one half" of the already smallest-possible spin 1 was like trying to visualize one-half of a skip rope loop. After all, in a skip rope you can have one loop, or two, or even more with expert skip rope twirlers -- but less than one loop? What does that even mean?
So just how mysterious is this half-unit of spin?
Reluctantly at First, He Took it For a Spin
Well, Wolfgang Pauli was easily one of the most brilliant (and abrasive) members of the very elite club of physicists who in mid 1920s developed the foundations of modern quantum mechanics. Pauli at first rejected even the idea that point-like electrons could spin, and likely cost Ralph Kronig a Nobel prize because of it. Pauli chastising Kronig so severely just for bringing up the idea that Kronig thereafter argued adamantly against his own idea! Pauli on the other hand subsequently not only repented of his initial view, but ended up developing the mathematical model for spin $\frac{1}{2}$ that is used to this day. The model is called the Pauli spin matrices.
But even someone as intimate with the issue of spin as Pauli pretty much gave up on any kind of conventional explanation of it. Instead, he declared particle spin to be an "abstract property" (p.3, line 9 from bottom) that has no particular connection to ordinary rotation. However, since quantum spin is a just a quantized version of everyday rotation, it is unavoidably deeply linked to it. Thus a more accurate translation of the word "abstract" in this particular context might be: "The math works beautifully, so please just use it and stop asking me what it means!"
So in summary, matter (which mostly likes to stay put, has volume, and resists compression) is built up from fermions whose rotations all have odd spin $\frac{1}{2}$ offsets in their rotations, while energy (which most often is literally as fluid and ephemeral as light and sound, and which can be compressed or focused almost without limit into a small volume) is built up from bosons whose rotations are all multiples of spin 1.
Lies, Darned Lies, and Spin Statistics
The spin statistics theorem is the formal name for all of that, stating that particles with spin $\frac{1}{2}$ are subject to Pauli Exclusion ("volume"), while particles with simple integer spin (or zero spin) are not subject to it. This theorem is primarily a summary of experimental findings; it is not some kind of mathematical result from which fermions and bosons were predicted based on first principles.
And that is why the connection between volume -- the resistance of matter particles to being compressed -- and the spin $\frac{1}{2}$ offset of fermions remains more a mystery than well-understood principle of physics. The proofs devised for it remain unconvincing even to the experts. For example, a 1998 assessment of spin statistics theory by Ian Duck and E.C.G. Sudarshan provides a detailed summary of the strategies theorists have used in trying to prove the spin statistics theorem, yet it concludes with this final line:

"Finally we are forced to conclude that although the Spin Statistics Theorem is simply stated, it is by no means simply understood or simply proved."

Two examples of such proofs include a very early (and still persuasive) [proof by Julian Schwinger, and this much more recent 2003 theory by Paul O'Hara. 
Invisible Hands, But Not the Adam Smith Type
One reason why I do not find any of these proofs particularly persuasive is this: If the theorists who created them did not know in advance exactly where they needed to go, it appears unlikely they ever would have managed to arrive at their destination. That situation is in sharp contrast to Paul Dirac's Dirac equation, which remains the gold standard for experimentally predictive theoretical mathematics. Once he came up with it, the Dirac equation pretty much had to drag Dirac kicking and screaming into acknowledging that there must be an entire second universe of antiparticles that are mirror images of regular particles.
The Conclusion
So, while various methods used to prove the spin statistics theorem may well be correct, they feel more like someone forging a circuitous path through deep woods to make their way finally to a bright light they could see off in the distance at all times. It seems quite likely that the main road, the easy path that shows you exactly where that destination lies, has yet to be uncovered. A truly simple explanation of why spin $\frac{1}{2}$ offsets lead to Pauli exclusion, and to its simple, everyday consequence that two objects cannot occupy the same space at the same time, has yet to be found.
A: The short answer is no. The slightly longer answer is no, but More is Different.
No, physics explains all everyday phenomena
There is act least one interpretation in which Michelson's statements were actually correct when he said them, and would be correct today. This is the interpretation given by Sean Carroll (thanks to @Michael Brown for the link). Relativity and Quantum Mechanics have enormously changed the way we look at the world, but the laws of physics that Michelson knew are still correct in the appropriate regimes. Compared to his everyday experience, relativity and quantum mechanics are "just" corrections in the "sixth decimal place."
Historically, special relativity was needed because of the negative result of the Michelson-Morley experiment. That negative result was meaningful only because Michelson could measure the speed of light so much more precisely than anyone else in the world at the time. His experiments were trying to measure variations in the speed of light; no one else would have been able to measure a change in the speed of light caused by the motion of the earth. He describes in my copy of Studies in Optics that the motion of the earth in its orbit would, under the aether theory, change the measured speed of light by one part in 10,000; he saw no change to one part in 400,000. My point here is that all the new physics that has been discovered since Michelson fits into the regions that we hadn't been able to explore until after Michelson. 
A good modern example is the fine structure constant $\alpha$. $\alpha$ is known to one part in $3.2\times 10^{10}$. When we discover new physics, it will agree with current physics in the first nine digits of $\alpha$. If it doesn't, then it's wrong. Any physics that isn't already known has to fit within the 10th digit of $\alpha$ and beyond.
But More is Different!
So that's why you could say that physics "explains" all everyday phenomena. But saying that physics explains everything, and that we understand the physics, doesn't mean that we understand all everyday phenomena. I can think of two good examples.
Protein folding is tangentially related to my work. Proteins are a bunch of amino acids covalently bonded to each other in a chain, possibly with some branching. The amino acids are in turn a bunch of atoms covalently bonded to each other. Physics explains covalent bonds: there is an electrostatic interaction, and quantum mechanics tells us what kinds of bonds are allowed given that particular interaction. In addition to that, each amino acid along the chain has a net charge that depends on the pH of the environment. How the protein chain prefers to fold itself up is then a thermodynamics problem. All of these electrostatic interactions within the chain, between chains, and between the chain and its environment add up to give the total energy of a particular configuration. But different configurations also have differing entropies. So how the protein folds up is fully described by physics as a competition between the entropy of each possible configuration and the energy of that configuration. But knowing the physics of how it works is a long way from being able to predict how it should happen, and when it will fail.
This counts as an everyday phenomenon because incorrectly folded proteins are believed to be at least an intermediate cause of several human diseases (most notably Alzheimer's), and because cooking food unfolds ("denatures") the proteins in the food. 
The other example I would give is consciousness. Again, we understand that the human brain is materially made up of cells that interact with each other by electrical pulses. But we are a long way from understanding how that physics gives rise to the biological phenomena.
I would say that these examples are physics in at least one important sense. It is somewhat arbitrary where we draw the lines among physics, chemistry, and biology. But we know they all have to be consistent with each other. Chemistry describes the stuff that biology takes for granted, and physics describes the stuff that chemistry takes for granted. One of the main points of the More is Different paper is that, at a certain point, it is much more useful and productive to stop worrying about the physics, just think about the chemistry or biology, and try to describe it in its own language.
A: It depends on exactly what you consider 'everyday' ... for instance, I work with solar physicists, and some of them work with attempting to understand the sun so that they can predict when it will launch CMEs or flares, and what causes the changes in activity through a solar cycle.  The sun doesn't launch CMEs every day, though.
If you wanted to build a list, I'd suggest looking at the (U.S.) National Research Council's various 'decadal surveys' for each discipline.  (eg, for heliophysics), which includes information about what type of research should be prioritized for funding ... and that basically comes from which questions the community is still trying to answer.  Also see the various whitepapers that were taken as input (eg, for heliophysics again), as that would cast a wider net.
... and those are just the questions that we have right now.  As we make new major breakthroughs (eg, quantum physics), we have new ways of looking at the world that show that there are a number of processes that we've just never considered before.
A: The flow of a fluid, such as air, is something very common place. It is what provides lift and drag on airplanes. You can feel its effects if you stick your hand out the window of a moving car. However, we still do not know everything about the state of fluid flow called turbulence and how a laminar flow transitions to it.
Although the governing equations for continuum fluid flow are well known (Navier-Stokes equations), the theoretical understanding of their solutions is incomplete. It is possible to directly solve the Navier-Stokes equations using numerical methods, however, this is very time consuming and not practical for high Reynolds numbers due to the wide range of scales.
Most engineering CFD codes solve the RANS equations instead which require a turbulence model to represent the Reynolds stress. AFAIK, there is no turbulence model that accurately models the flow in all situations. (especially for separated flows)
Turbulence is a physical phenomenon that most people see daily, even if they are unaware of it. You can see it in water coming out of a faucet, clouds, stirring cream in coffee, etc, but the theoretical understanding and modeling of turbulence is far from complete.
A: Consciousness remains to be explained by physics. The emergence of free will (apparent or not) remains to be explained by physics. 
In short, the everyday phenomenon referred to as 'life' remains to be explained by physics.
A: Sonoluminescense is an interesting one.
A: Any problem with multi-body interactions has not really been explained. By that I mean anything which cannot be explained as an ideal gas, ideal of excitations (e.g., photon, phonons), or has well-defined periodicity (so it is linear in frequency space).
This pretty much includes most liquids/glassy states, turbulence, earthquakes, emergence of order from underlying disorder (wind patterns, stripes on zebra),...., life
We do know many of the properties of these systems, i.e., chaotic behaviour, emergent macroscopic patterns and such, but we are not even close to any explanation. I think there is a lot more to be "understood" before it can be computed.
A: A critical observer might say we don't even understand the motion of a falling object (or indeed any classical motion) in a rigorous way. Indeed, classical mechanics is built on the 'Principle of Least Action'. This principle actually has quite an inaccurate name. A more appropriate name would be 'Principle of Extremal Action'. Indeed, the Euler-Lagrange equations follow from the condition that the variation of the action as a consequence of a variation of the path vanishes, or in other words: the actual path nature 'elects' extremizes the action.
So nature does not elect the path which minimizes the action (we could understand that), rather it chooses a path which eiter minimizes or maximizes the action. This sounds incredible: nature taking the path which maximizes action? Why? We don't know... Yet the entire theory of classical mechanics is built on this. Obviously, classical mechanics works and it's all fine, but we do not understand why it works the way it does. Not really. I personally found this incredible when my professor told it to me.
A: RIEMANN ZEROS and QUANTUM CHAOS :) the similitude between these 2 apparent phenomena : Numer theory and physics still remain unexplained
