First you'll have to install the Git command line tool on your machine, following these instructions. Then find the repository that you want to contribute to, copy its address from the green "Clone or Download" button, and on your local machine run e.g.
git clone https://github.com/dmurfet/difflinearlogic.git
To see a list of what has changed (optional) run git status. Then
git add *
git commit -m "Changes"
git push origin master
Instead of * you can manually specify the files to be added to this commit. You're also encouraged to use a more helpful commit message than just "Changes".
Make sure you have committed all local changes before fetching the updates.
git fetch
git merge origin/master
Usually this isn't a big deal. Edit the file to manually resolve the conflict, re-add and re-commit. See this guide.
Your workflow should look like this: when you sit down, fetch changes from the server. Make your edits, regularly committing the changes and pushing them to the server. When you've finished, make sure you commit and push one last time. For further help see the GitHub guides.