Why can't an electron be observed? I was watching a show on Netflix hosted by Neil Degrasse Tyson and he mentioned that one of the fundamental particles that we know of, the electron, is something we have never even observed directly. Why haven't we done so? Is it impossible? I know this is the easy answer to any question, but is it because our tech isn't advanced enough yet?
 A: The concept of "direct observation" is a tricky one in the philosophy of science.  It is the accepted bridge between scientific theory and truth, and it goes over some very muddy waters.
Consider this profound statement: we have no proof electrons exist.  Zero.  Nada.  What we do have are many theoretical models which include the concept of an electron which do a remarkably good job of predicting how things will behave.  In philosophy, this is the divide between ontology (the discussion of what the world is) and epistemology (the discussion of what we can know about the world).
Now this is probably the most pedantic viewpoint one can choose.  You'll almost never hear a scientist choose to talk this way.  Why?  Well some of our models do such a mindblowing good job of predicting things that we tend to like to just claim they're "reality."
How do we make this claim?  If the model predicts something which we can "directly observe" using our own eyes ears and hands, then we presume it is "real."  The idea that single celled organisms make food go bad was just an indirect observation until someone invented the microscope and let us look with our own two eyeballs.  At the philosophical level, we "bless" observations made with our own senses, for no reason other than that its really hard to make any progress if we don't trust anything.
Take the idea of bending spacetime.  We've all heard of Einstein's theory that mass bends spacetime, and that causes gravity plus all sorts of other fun relativistic effects.  However, bending spacetime is only a model.  There's no proof that spacetime actually bends, just that if one models spacetime as bending, one gets characteristically good predictions about what will happen.
So we have no blessed microscope which can magnify enough to see an electron.  That on its own is enough to make one state that no one has "directly observed" an electron.  If one dare dabble into Quantum Mechanics, the world gets even stranger.  Due to all sorts of funny effects which are probably beyond the scope of your question, the concept of "observing an electron" gets rather muddy itself.  Quantum Mechanics predicts what will happen in the universe in a way that makes the phrase "observing an electron" troublesome and difficult to really quantify.  If indeed Quantum Mechanics describes how the world "really works," then the concept of observing an electron may actually be impossible due to the statistical behaviors of quantum waveform collapse.
A: Nothing is observed without some physical process 'performing' the observation. Eyes don't passively see - they are huge collision sites for countless photon bombardments, camera pixels (or photosensitive pigments) don't passively react to light, they have to be violently pounded by thousands of individual photons. 
It's like being in a car, in a dark room full of feathers - how can you, without leaving the car, sense that the feathers are there? If you drove into the cloud of feathers at 200mph, you might hear something as they struck the windscreen, if you listened very carefully...
It's the same down at individual particles such as an electron - how can you sense something that is lighter and more delicate than anything else around it? 
Like the car and the feathers, only by bashing one thing into the other violently enough can you indirectly infer one, or the other's existence.
A: Define "observe"
The Webster, the options for science are:

a: to watch carefully especially with attention to details or behavior for the purpose of arriving at a judgment
b: to make a scientific observation on or of

And on "observation:

a: an act of recognizing and noting a fact or occurrence often involving measurement with instruments 
b: a record or description so obtained

How do we observe a cloud? Light from the sun scatters off the H2O of the cloud and hits our eye. To observe it scientifically one takes pictures, or videos to study its time evolution. Is the cloud on the video "observing a cloud"?
Here is a bubble chamber picture of an electron:

The electron has scattered off atoms in the chamber ionizing them and the bubbles are formed where the ions were. It is turning in the magnetic field imposed and is losing energy from the scatters. In contrast to the cloud picture, it is not light that scatters off the object, but the object scatters off matter, and light of that path is recorded. It is a more complicated path to a picture, but there is still a one to one correspondence of the object called electron, with the picture which we call observation.
Please note that for the dimensions of the bubble chamber bubbles (microns) and the momentum of the electron in the picture, some MeV/c, satisfy the Heisenberg Uncertainty Principle, and thus within these dimension the electron can be considered in its form as a quantum mechanical "particle", with attributes of a classical particle.
We also "see" electrons directly in sparks, our eyes and brain are not equipped to see the light as clearly as the light reflected from the cloud, but this is a limitation of our biology, our instruments can.
So I think that the statement is vacuous.
Edit after comment

Do you think it would ever be possible to make something that works like a camera to take a snapshot of a single electron

Searched the net and it has been done.

video at https://www.youtube.com/watch?v=OErXAk42MXU

Now it is possible to see a movie of an electron. The movie shows how an electron rides on a light wave after just having been pulled away from an atom. This is the first time an electron has ever been filmed, and the results are presented in the latest issue of Physical Review Letters.

So it has been done, though the video is slowed so one can see the path.
A: Let me give a vague analogy to illustrate the issues in talking about observation.
Imagine that I have just thrown a stone into a pond and I ask you, can you see the wave it makes? You say, yes of course, why? I say, no you didn't really see all of the wave. You only saw what you saw from your position and angle. And is what you saw the wave? Or is it just an image of the wave that is in your mind? How do you even know that that image is an accurate reflection of the actual wave? In fact we know our eye has a limit on its resolving power and its sensitivity.
So you didn't actually see any wave. You just saw something which both of us do call a "wave" in English. The word is merely a reference to the entity, not the entity itself. You might ask, is it possible to observe an entity directly and not through any intermediate instrument such as our eyeball? But what really is "directly"? If your mind can somehow touch the wave (with what, may I ask?), is it enough for you? Or is your mind itself merely an instrument which you use to interact with the world?
Anyway that dives straight into the deep end of philosophy, though you'll have to somehow answer that before you can specify precisely enough what you mean by "observe".
On the other hand, what if we both agree there was a wave, and I then ask you, what is the position of the wave? And you stare blankly at me. But that might well be the same question you would be tempted to ask about an electron. What if the electron indeed has an underlying reality that corresponds more closely to its wavefunction rather than a single point in space? Do you even have the slightest evidence that it is more like a point? No.
You could say, let's take the highest point of a water molecule (assuming it's sufficiently point-like) as the position. If so, then it's not going to have any nice properties at all, and would randomly jump around the pond. A better idea would be to take the average position of the water molecules that are above the average water level of the pond. Then we can 'see' that it moves in the direction of the wave more or less along with the crest. We can even touch the wave crest as it goes past, which means that we can sort of estimate the position as defined this way. There is some uncertainty here, not unlike the uncertainty you see in measuring the position of a classical wave packet or in measuring the position of a particle (Heisenberg's uncertainty principle), in the sense that despite the position of a pond wave being well-defined (assuming point-like water molecules or more generally some mass density function for the water), we cannot even classically measure it accurately because anything we do will disrupt the wave.
Similarly, we can define velocity of the wave as the flux, the average flow of the water (according to the evolution of the mass density function over time). As with position, we cannot even classically measure the velocity accurately without changing it.
Now, we could go another way around the problem. Instead of trying to observe all of the wave at once, we repeat the stone throw many times, and each time we observe just one small part of the wave. Of course, now that we are skeptical we'll question whether we really can repeat something in exactly the same way each time. It's of course impossible in general but we hope it's not too wildly different.
This is exactly what scientists have done in order to observe (in this sense) the wavefunction of an electron. It was done very long time ago, and I do not know the history, but supposedly IBM was one of the first to arrange impurity molecules on a metal surface and then using a Scanning Tunneling Microscope to image the electron density. They have some pictures here, including the well-known quantum corral:
(http://researcher.watson.ibm.com/researcher/files/us-flinte/stm16.jpg)

I do not know whether they edited the raw data (highly likely, when I did it before I had to edit to remove noise and artifacts from the imperfect STM tip). There are other images on the internet like:
(http://nisenet.org/catalog/media/scientific_image_-_quantum_corral_top_view)

But of course all colour or 3d effects in STM images are computer generated. Recently (2013), some have claimed to be able to image atomic orbitals, such as:
(http://physicsworld.com/cws/article/news/2013/may/23/quantum-microscope-peers-into-the-hydrogen-atom)

Anyway remember that analogies break down, and very little about electrons and other particles even has a vaguely analogous phenomenon with water in a pond. The analogy was merely to get you to think twice about the common assumption that a particle has a point position.
A: According to the Standard Model, the electron has no extent; a radius of zero.  As such, such a particle could never be observed (as it is not really there...) but only indirectly observed by, for example, the effect of its electric field on other particles or objects.
A: It isn't entirely accurate that it cannot be observed. It can, and has. However, observation is only a small aspect of the phenomenon of electrons.
While it can be observed instantaneously, it cannot be determined both where it is and its velocity. The speed of an electron can be observed, but not with knowing its position. The position of an electron can be observed, but not with knowing its velocity.
The often shown example of this in engineering and physics classes in the image of an arrow in the forest. You can clearly see where it is, but from an image of the arrow, you cannot tell how fast it is going.
A: According to relational QM, another observer of an electron could simply be another electron; say another electron that repulsed it.
This means observations are being made of electrons all the time; just not by us. 
But this, I think, isn't the sense of observing you're using in your question; which appears to be the direct observation by the human eye.
However, once we could not see bacterium; and now we can see them through a microscope; no-one says that's they're not there.
Perhaps the same will go for electrons, one day.
