Do electromagnetic waves have endpoints? When learning about electromagnetic waves at school we never talked about any endpoints as we did with standing waves, so I assumed that light has an endless length, but that doesn't make sense. So my question is, does it make sense for em waves to have specific length? If yes, how does it differ from one to another?
 A: Standing waves on a string is just one of many examples of transverse waves! If you have seen pictures showing electric and magnetic fields oscillating perpendicularly, then you might come to the conclusion that they are like vibrating strings. The fact that must be made clear is that, there is no material medium needed for the EM Radiation to exist, unlike strings. This should tell you that, we must look at EM waves as distribution of energy in space rather than something oscillating(like a string), hence rendering the question of endpoints as non-existential. However, you can trap EM radiation in structures called as cavity resonators: http://en.wikipedia.org/wiki/Resonator#Cavity_resonators
or Waveguides: http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)
A: The simple answer is by having a spread of frequencies / wavelengths in the EM field's spectrum it can be confined to a finite region. A perfectly monochromatic wave (i.e. one containing only one wavelength), whose electric field vector varies like $\cos(k\,z-\omega\,t)$ where $k = 2\pi/\lambda$ is the wavenumber and $\omega = 2 \pi f$ is the angular frequency has no "beginning or end" and would hold infinite energy. So the perfectly monochromatic wave is a theoretical abstraction used to simplify discussion.
Actually, this idea holds for all waves, even those standing waves on springs that you have experimented with. You actually have a forwards running and a backwards running wave - a mix of a positive wavelength and a negative one of exactly the same frequency magnitude such that the interference nulls the wave out at its endpoints, which are tethered to the experimental setup showing the standing wave. So a wave bouncing off two endpoints has a forward and backward component and looks the same between the endpoints as an infinite, lone-frequency wave with two components of equal amplitude running in opposite direction would look between the endpoint. So the fact of the nodes lets one kind of "excise" a finite section of the standing wave and keep it running. From the standpoint inside the cavity, there is no difference between the situation with tethered endpoints and one with two infinite waves. 
You can do exactly the same thing with light in a resonant cavity. This is indeed what happens in a laser, which sets up a standing wave between two mirrors. You can also see this kind of thing happening if you stand between two mirrors and see a theoretically infinite number of copies of yourself. By dint of the virtual images, there is no difference between the light field making your image between the mirrors to a lightfield of infinite extent bearing an infinite number of images of you!
More generally, you think about how real waves have end points by thinking about Fourier transforms, which you will learn about after high school. But the basic idea is simple enough. First of all, any periodic function $f(t)$ such that $f(t) = f(t + T)$ where $T$ is the period can be "built" as a sum of functions of the form $\cos(2\pi\,n\, t/T)$ and $\sin(2\pi\,n\, t/T)$, where $n = 0, 1, 2, 3, \cdots$. So it can be built of sines and cosines with the same frequency as well as their higher harmonics. This is a Fourier series. To go forward to the Fourier transform, we simply let the period $T$ get bigger and bigger so that in the limit it comprises the whole real line and thus any reasonable function can be built of monochromatic waves with a spread of periods. To build a short pulse, the Fourier transform adds together a wide frequency spread, so that there is strong destructive interference between the monochromatic components aside from near where they are all perfectly lined up in phase, i.e. near the pulse. 
The shorter the pulse, the wider the spread of frequencies needed.
This may sound familiar from your quantum mechanics research: it is the underlying mechanism of the uncertainty principle. It turns out that to express a function in momentum co-ordinates, you take the function expressed in position co-ordinates and take its Fourier transform. And contrawise. So if the function is perfectly monochromatic - of the form $\cos(\omega t)$$ - so it has perfectly known momentum, it is uniformly spread over with real line and its location in position co-ordinates is altogether uncertain. The formal statement of the uncertainty principle follows from the "wide spread in frequency means a narrow spread in time/space and contrawise" property of the Fourier transform.
A: As pointed out in the other answers, electromagnetic waves can be spatially confined in a variety of ways, but perhaps more fundamental is how electromagnetic waves can be confined temporally. 
Electromagnetic radiation is changes in the electromagnetic field propagating through space. If you imagine a point charge emanating electric field lines, then when that point charge moves to another location the electric field lines must now emanate from that location. However, information about the new configuration of the field can only travel at the speed of light, so the motion of the point charge translates into a disturbance in the field that propagates away from the point charge at the speed of light.
This disturbance is an electromagnetic wave, electromagnetic radiation.
Therefore, the endpoints of such a wave are defined by the beginning and the end of the propagating disturbance. If the charge only moves once, the wave will be very short. If the charge is attached to a motor the pushes it back and forth indefinitely, then the wave will have an end point at wherever the wave front of the disturbance is, and it will also have a sort of end point at the charge itself (which produces the wave by constantly moving its location and thus changing the nearby electromagnetic field).
