Most colleges, universities, academies, etc. have their own requirements to apply. Make sure to look into the specific schools and programs that you're looking in to.
The art schools in the US are generally normal colleges and universities:
And so the requirements for those will generally be high school degree/transcript (or GED), standardized test scores, letters of recommendation, and art specifically will generally require a portfolio of your best work. Since you'll be enrolling in a school you won't need to be a master artist before you go in, but you will need to demonstrate passion and some fundamentals.
If you're looking into the more vocational schools like Ex'pressions, Full Sail, Academy of Art, etc. keep in mind that many of those are for-profit institutions. And while they do provide classes and degree programs, be aware that they might not be quite as focused on education as much as tuition. That's not to say that you can't still get a quality education from those places, but you should be aware that they answer to their stock holders more than their students. I personally attended Academy of Art University in San Francisco and I learned a lot there.
But they're not for everyone and they do come with some controversy in terms of how they handle federal grant money, student loans, professional placement, etc. Just make sure to do your research when looking at places.
I hope that this answered your question. Good luck in pursuing your degree!
A good place to start with training is at your local community college. Go to the Alumni Relations officer of the community college, let you know the field that you are contemplating, ask to be put in contact with alumni that have the appropriate degree from the school or a job that looks interesting to you. Go and talk to them at their place or work to get a feel for what they do and where they do it.
Go to the reference librarian at you local library and ask to be put in contact with a person in a professional organization in your area which represents the type of job that you are considering. Contact them and arrange to go to a meeting as a guest and do networking.
Look in the phone book for any organizations in your area that might do the kind of work that you might want to do. Go to them and find out what type of training required and where they might recommend getting that training.
Do not be afraid to ask. People are very willing to help people who are doing career exploration.
Also, tell every one that you meet what your interests are and ask them information. Help can come from unexpected sources at unexpected time.
Do not forget to send thank you notes to people that you talk to.
I'm not sure if you are a high school student or a college student looking for post graduate training. When you say "institute" I take it you mean an art school, college or university that offers a program in art and animation?
As a previous volunteer mentioned, each school has its own requirements for admission. There are hundreds of schools that offer programs in art and in animation. I would talk to the art teacher(s) at your current school and see what they suggest, or where their previous students have studies and so on.
Whatever the specifics, you will have to show that you are a good student, that you are willing to learn and to work hard. This will be reflected in your grades for your general ed classes and in your letters of recommendation. You will have to have a portfolio of your artwork that shows not just your best work, but a range of art materials and approaches. Computer graphics are becoming a mainstay of art departments everywhere, so familiarity with Photoshop and others will be important. This is especially true it you plan to go into animation--get some experience with the basic software programs; take some CG classes. Again, each school will have their own portfolio requirements, so start looking at schools and narrowing down your choices.
The question really has a more appropriate beginning: What are you really interested in and what sort of education do you really need to do it?
Start by going to the Alumni Relations officer of the school that you recently graduated from. (perhaps it was high school) Have them put you in contact with people who are doing the type of work that you might find interesting. Go and visit them at their place of work. Find out what they do and what education they find necessary and where they recommend you go. Look in the phone book and find organizations doing what you want to do and visit them.
You might be able to get the type of education that you are seeking from your local community college as a much cheaper rate. Go to the Alumni Relations officer and arrange to talk with alumni in jobs that look interesting to you.
The most important thing to do is first find out what you really want to do and find the shortest and most economical way to get there. You do no want to be burdened with student loans.
Do not forget to send thank you notes to people you have talked with.
Also, let everyone you know or meet about your interests. Helpful information can come from surprising sources and surprising time.