You can do this in one of 2 ways –
Direct the user to the below address. This will direct the user to a page that has a token that she can copy and paste back to you. The important bit is that you ask for expiration = never and scope = read,write
https://trello.com/1/authorize?key=substitutewithyourapplicationkey&scope=read%2Cwrite&name=My+Application&expiration=never&response_type=token
Or use OAuth (harder) to automate the request for an access token. Read more in the documentation.
Once you have the token, you can make any API call you’d like.