How do stars from far away affect Earth? I know that we obviously get light (or we wouldn't be able to see them), but are there any other ways that they affect Earth and maybe just our solar system in general?
 A: The stars in our galactic neighbourhood do have a dynamical, gravitational effect on the inner workings of the solar system:
They built the Oort cloud
The Oort cloud is a roughly spherical cloud of icy bodies that is thought to act as a reservoir of long-period comets (and which we speculate exists to explain said comets' existence). These icy bodies formed in much the same way that Kuiper belt objects did, accreting in circular orbits on the outer edges of the solar system, but then they had gravitational interactions with the planets that scattered them into higher orbits.
What happened next is probably best explained by Hal Levison in this interesting interview with Emily Lakdawalla:

You start out with a bunch of icy guys between the planets. And they start to scatter outward. If you think about it, you do planet flybys to get velocity kicks. Spacecraft do it all the time. The planet can't put you on an orbit that doesn't cross its orbit anymore, because it's just a velocity kick, it's not a position kick. So in the early stages, when the planets are forming, they're scattering these cometary objects around in such a way that the perihelion still remains among the planets, but the semimajor axis is doing a random walk as you get scattered. So the semimajor axis is bopping around and slowly growing because it's diffusing outward, until the point where the galaxy can become important gravitationally. And that acts as a torque. What the galaxy does is it doesn't change the semimajor axis, but it does change the perihelia distance. So you get out to a few thousand AU and the galaxy can lift the perihelia away from the planetary system and then you're frozen in the Oort cloud.

The orbits of the Oort cloud objects are huge because they were boosted there by gravity assists by the planets, but as Levison notes this process can only produce very tall ellipses that still intersect the planet's orbit - at the location of the original assist, because all elliptical orbits are closed.
The galaxy, on the other hand, is too far away to lift anyone's orbit up - it cannot provide orbital energy - but what it does is circularize these long, thin elliptical orbits. This is what Levison refers to as a torque: trading radial kinetic energy into angular velocity. This makes the orbit less elliptical while keeping the semimajor axis constant (because this keeps the energy constant), and it therefore pushes the perihelion away from the 'inner' solar system (i.e. from Neptune inward) until the object can no longer meaningfully interact with any planets, at which point the size of its orbit is sealed, and the only possible change that can happen is further drift of its ellipticity induced by extra-solar stars.
That feels like enough to be getting on with to me.
A: The other answers talk about some of the effects.  This is a complementary answer that attempts to put a number to the force behind one of the effects - gravitational attraction.
Proxima Centuri is the closest star to our solar system.  It is about 4 × 1016 m away and has a mass of 2.45 × 1029 kg.  The mass of Earth is about 5.97 × 1024 kg.  Plugging these values into the standard gravitation formula, we find the force of gravitational attraction between Earth and Proxima Centuri is:
$F = \frac{G m_1 m_2}{r^2}$
$= \frac{6.67408 × 10^{-11} × 2.45 × 10^{29} × 5.97 × 10^{24}}{ (4 × 10^{16})^2 }$
$= 6.1 × 10^{10} N$
This is larger1 than I had expected when I started this calculation.
Obviously this force is not acting on the Earth in isolation.  Pretty much all objects of any significance in our own solar system exert vastly greater gravitational forces than this on the Earth whose effects should be considered before any potential effect from Proxima Centuri.
In addition all solar system objects will have the about the same force (proportional to their mass) exerted on them by Proxima Centuri and thus the solar system will act more or less as one object with respect to Proxima Centuri in this regard.  There is certainly no expectation that the Earth (or any other solar system object) will suddenly pop out of its normal motion and zoom off in the direction of Proxima Centuri.
Some interesting comparisons have popped up in the comments;  I'm tabulating them here for the record (figures are all very approximate):

$6.6 × 10^{9} N$ - Weight (at sea level) of the largest ever ship (Seawise Giant), fully displaced.
$6.1 × 10^{10} N$ - Gravitational force of attraction between Earth and Proxima Centuri
$6 × 10^{11} N$ - Gravitational force of attraction between Earth and Pluto
$2 × 10^{20} N$ - Gravitational force of attraction between Earth and the moon


Note also that the combined mass of the binary Alpha Centuri is a bit more than 16x the mass of Proxima Centuri, and is barely (in astronomical terms) any further away.  So its force of gravitational attraction to solar system objects will be 16x greater.

1 Please comment/edit if you see an error in the calculation.
A: I don't think that light from the stars other than Sun is of much practical use nowadays except for the classic navigation, where it's essential of course.
I guess any effect comes from the limitless reach of the gravitational force, which drops with the square of the distance but grows linearly with the mass exerting the force. A star most obviously affects planets in its system. A bunch of stars are affecting their star cluster, and so on.
Check this subchapter in the Feynman's lectures.
A: Gravitationally, there is little immediate effect on earth on a daily basis, though over very long periods of time, stars that pass near enough to the sun could disrupt the orbits of Oort cloud objects and send them towards the sun (and earth or other planets in our Solar System).
Culturally, stars have a very big impact on our species. Religion, art, science, and other areas of our culture draw inspiration and knowledge from observing the stars. As we study and understand more about the stars and their planets, we understand more about ourselves and our place in the universe. Arguably, we would not have the same culture without having the stars with us over the centuries, and we would have had a different impact on the earth than we have had. Because the stars affect us and we affect the earth, the stars indirectly affect the earth through their effects on us.
A: *

*The gravitation of the stars in our galaxy keep the solar system in it. I'm not sure whether that's important for earth or for life on earth though, but it makes for nicer night skies.

*Gamma ray bursts, if close enough and (im)properly oriented, affect earth. A gamma ray event from a "soft gamma repeater", SGR 1900+14,  is known to have affected earth's atmosphere, although it was 20,000 light years away.
The stronger events are so powerful that a close one would be devastating. Luckily they are rare (a few per million years per galaxy).
A: A lot (to put it mildly) of elements are created in stars and supernovae. These elements then travel through space until they fall to Earth (or, to be exact, some microscopic portion of them reach us).  Earth itself wouldn't exist if stars hadn't generated elements which then clumped into dust, into minerals, and so on until a big ball of matter started to orbit the Sun.
Here's a short quote from Wikipedia article on Cosmic ray:

Data from the Fermi space telescope (2013) has been interpreted as
  evidence that a significant fraction of primary cosmic rays originate
  from the supernovae of massive stars. However, this is not thought
  to be their only source. Active galactic nuclei probably also produce
  cosmic rays.

So I'll stand by my claim that stars are giving us mass (i.e. non-photons) as well as photons in real-time, not just as 5-billion-year-old space dust. 
A: Machs principle, that inertia is caused by the distribution of distant stars was a principle that Einstein tried to incorporate into GR, but failed.
However Barbour, quite recently incorporated an aspect of Machs principle into his theorising of time: ephemeris time

An ephemeris gives the position of celestial bodies, and duration is deduced in terms of such positions.

And

a man-made clock, a chronometer, is "a mechanism for measuring time that is nearly as possible synchronised with ephemeris time". 

Overall, the thrust of Barbours paper is to show that time is an aspect of a timeless law that governs change.
A: The distant stars are also responsible for cosmic rays, which in turn can affect the Earth's weather
