There are many free, downloadable, computer animation programs on the web.
Note - Many 2D programs have stopmotion capabilities.
Each category gives a short explanation of what type of animation the programs in it are for, and what you can do with them.
When relevant, I've added links to animation software which are not free, but are cheap or have a trial period, or a demo version.
Also, you will find in each category a list of the professional, industry standards.
Here is a thought to keep in mind about the paid-for programs versus the free ones:
There is a reason why a heavy, well rounded software became a standard, and that's usually a large range of abilities and a friendly interface.
Sometimes though, a sweet little freebie can do exactly the one thing that you need,
So define your needs first.
Don't be afraid to experiment and combine resources, but please use caution:
Check all files you download with an antivirus program, and don't ever give your details to suspicious looking prompts.
For my part, I will not publish links to places I don't trust, but obviously I cannot be responsible for other people's materials.
Have Fun!
You might also want to know:
2d animation software
vector based 2d cartoon animation software
3d computer animation software
stop motion animation software
From our home Test Pilot:
animation software review - MonkeyJam