If Git prompts you for a username and password every time you try to interact with GitHub, you're probably using the HTTPS clone URL for your repository.
Using an HTTPS remote URL has some advantages: it's easier to set up than SSH, and usually works through strict firewalls and proxies. However, it also prompts you to enter your GitHub credentials every time you pull or push a repository.
You can configure Git to store your password for you. If you'd like to set that up, read all about setting up password caching.