What OS do you have? Also, what style are you going for?
Without knowing either of those, I will list some that I know of and/or have used. I suggest that you download at least a few of these (or, for the non-free ones, download the trial) and try them out. One person's favorite graphic editor may be another's worst nightmare (really!).
If you're on Windows, I recommend GraphicsGale for pixel art and pixel art (i.e. non-Flash and non-skeleton) animation. The free version should have anything you should really need for game dev. This is my personal favorite for pixel art
http://www.humanbalance.net/gale/us/
For Mac, I've heard Pixen is a good pixel art editor:
http://pixenapp.com/
If you don't mind Adobe Air, Pyxel Edit is for Mac and Windows and is really good for making tiles:
http://pyxeledit.com/
I've heard some good stuff about grafx2, but I have never tried it. It's another pixel-art-centered one. It's for Windows and Linux, with (apparently) unofficial support for Mac OSX:
http://code.google.com/p/grafx2/
EDIT: For another good pixel art program, see my edit at the end of this post
Spriter is a good animation program for Windows, Mac, and Linux:
http://www.brashmonkey.com/spriter.htm
For animation and vector art, Flash Professional is always a good option, though expensive:
http://www.adobe.com/products/flash.html
Anime Studio is good for Flash and 2D vector skeleton-based art and animation too:
http://anime.smithmicro.com/
For more detailed art, there's always GIMP (Windows, Linux, and Mac) and Adove Photoshop (Windows and Mac), with the former being free and the latter being expensive (though if you're a student then you should be able to get it pretty cheaply).
You can of course use both for pixel art, but many find them to be a bit overkill and/or too cluttered for that (but many don't and use them for pixel art just fine).
http://www.gimp.org/
http://www.adobe.com/products/photoshop.html
I've heard some good stuff about Paint.NET, though I have never used it. Free, Windows-only.
http://www.getpaint.net/
For vector art (with no or little built-in animation features, at least without plugins [?]), the two most popular options are probably Inkscape and Adobe Illustrator. Like GIMP vs Photoshop, the former is free and the latter is expensive. Unlike GIMP vs Photoshop, Illustrator imho is far, far better. I personally prefer the Adobe offerings in both cases, but the gap between Illustrator and Inkscape is much larger than the gap between Photoshop and GIMP (ESPECIALLY regarding the interface). Inkscape is for Windows, Mac, and Linux, and Illustrator is for Windows and Mac.
http://inkscape.org/
http://www.adobe.com/products/illustrat ... moid=KAUCB
And, here are a few others (some Windows-only, some Mac-only, some for Windows and Mac, some for Windows, Mac, and Linux):
http://www.thebest3d.com/dogwaffle/
http://www.artrage.com/
http://www.pixelmator.com/
http://www.cosmigo.com/promotion/index.php
http://www.corel.com/corel/product/inde ... rod4900067
http://www.pickleeditor.com/
http://ragepixel.com/
http://tilestudio.sourceforge.net/
http://www.flat2d.com/easytoon/telechargement.aspx
http://www.pencil-animation.org/
http://www.corel.com/corel/product/inde ... rod4030123
http://www.corel.com/corel/product/inde ... rod4950148
PS: If you're curious about what I like...I use Graphics Gale Free Edition, Photoshop, and Illustrator. I also use Pyxel Edit a little for tiles, but have yet to do anything substantial with it. Pickle (
http://www.pickleeditor.com/ ) and RagePixel (
http://ragepixel.com/ ) are two that I've
just come across and that really interest me - I sort of like the idea of a less Windows-y interface than GraphicsGale (that may seem like a silly aesthetic thing, but for me that matters when I'm in "creative mode").
EDIT: I've just tried Pickle (
http://www.pickleeditor.com/ ), and I REALLY like it! Very simple and easy to use, yet not devoid of features. So far, I recommend it!