As the days go on I'll talk about each piece of software! You need to know at the outset, though, that my favorite (both for ease of use, value, and price for value) is Toonboom Studio. If I were recommending a piece of software to a new animator it would be the one I'd recommend. Flash is okay for animation for the web, and Mirage is wonderful. The nice thing is that both can be used with Toonboom software! So my bias is Toonboom Studio. They also have academic versions at a very reasonable price, so if you're a student you can own a very powerful yet easy animation program for little money.

I am currently doing 2d animation using Toonboom Solo, Mirage, After Effects and Flash in combination.

Traditional and Cutout 2D Animation Software:

Toonboom Studio (the best!)
Toonboom Solo
Toonboom Studio Express
Manga Studio Ex
Manga Studio Debut
Comicbook Creator
Mirage (great!)
Flipbook (very traditional animation and a nice program)
Monkeyjam
PAP (shareware)
Anime Studio Pro (commercial Moho)
Moho (free)
Toonworks
Flash
Corel RAVE
Paint Shop Pro (minimal gif animation)

3D Software:

Blender (free and great!)
Maya 7 (now that Autocad bought them it'll be even more expensive, but full featured)
Anim8tor
Life Forms
Daz Studio
Shade
Poser
Bryce
Ray Dream
Hash Master

General drawing programs and others hard to classify that will do animation:

Gimp (free)
Adobe Photoshop
Adobe Illustrator
Corel Draw
Corel Painter
Adobe After Effects
Morpheus