I used to work in an art studio where they trained students to work in the video game industry for Warner Brothers and Ubisoft. The product they used most (and they used several) was Autodesk Maya. It isn't free, but if you are a student (I.e. registered in an accredited school) you can get it free for 3 years.
The classic free animation software is Blender, it's available for Windows, Linux, and Mac and it is very powerful, but it has a fairly steep learning curve. The most famous video produced with Blender has to be "Big Buck Bunny" and as you can see the results are truly professional. If you're on a low budget and are willing to put the time in to learn how to use the product then it would be an ideal choice.
I create animated explainers to use within all of my videos and I make them using Apple Keynote. It's not really animation software but for simple sequences it is amazing and, best of all, it's free. Of course, being an Apple product it only runs on the Mac. You can learn how to use it in about 2 hours and when you export your videos using a Prores 4444 codec the quality is excellent. But again, it is not really animation software, more like powerpoint on steroids.
Hope this helps!