Modifications to a project which is version-controlled with git are done through commits. In git you modify your project until you have reached a state you are satisfied with and then you commit all these changes together to the repository in one single commit.
Modifying the content
If you have a project which is already shared with git then doing the actual modifications is easy: just modify or delete files either with eclipse or even directly on the file-system. There is no need "check-out" files or to tell git in advance about these operations. Only when you add new files you should be cautious: new files which should be version-controlled have to be explicitly announced to git through the files context-menu.
When you are satisfied with the state of the project you may commit your recent changes. To do that select "Team -> Commit..." from the context menu of a project or a file in the project.
It does not play a role whether you select this action on a certain file on on the whole project. The reason for this is the following: in git you never commit individual changes to a certain file. Instead you always commit a new state of the whole git repository capturing the state of each and every version-controlled file in that repository. There is no commit on a certain file or even on a certain project. A commit will capture all eclipse projects which are configured to be version controlled in the same git repository as the project on which you initially triggerd the commit.
Once you have triggerd the commit the following dialog will pop-up.
In this dialog you specifiy the commit-message. This text may describe for example what you have changed and why you did it.
Additionally can control in this dialog which of the touched files will be included in the commit. If you clear the checkbox in front of a file, the changes to the file will not be included in the commit. The local file in your eclipse workspace will still contain the modifications giving you the chance to commit these changes with a subsequent commit. This feature is often used to seperate the modifications you did to a set of files into different commits.
One example: Imagine since the last commit you have fixed a bug in A.java and you have added a new method to B.java. This two modifications are independent from each other and you may regret that you have not committed directly after you have fixed the bug in A.java. In this case you initate the commit, deselect B.java from the set of committed files and specify a commit message describing only the bugfix in A.java. After a succesfull first commit you just call commit again and the upcoming dialog will present you the remaining changes in B.java. Now you can specify a commit message for the addition of the method and finish the commit.
Be aware that new files which you added to the project and which you have not explicitly added to version control (see "Modifying the content") will not be listed in the commit dialog and will not be part of the commit. If you are not sure whether you have created such files you may select "Team -> Add to Version Control" on an eclipse project to add new files in the project to version control.
When you commit you may specify that the current commit should "amend" the previous commit in the current branch. Your commit will then replace the previous commit. This feature is often use to correct wrong commits.
Example: Imagine you have committed a change to a file and after the commit you find out that the file contained a typo.
After you commited you detect the typo. In order to correct this typo and to get rid of the wrong commit you just fix the typo in the source file.
Afterwards you trigger the commit and select the option "Amend previous commit".
When you do so the commit message of your previous commit (the one which you want to replace) is filled into the "Commit Message" field. This gives you the chance not only to correct errors in the content of the version-controlled files but to also correct errors (e.g. typos) you did in the commit message.
As an alternative to amending previous commits you could just commit the corrected version as a subsequent commit. But the first commit containing the typo is of no use to anybody else and in order not to clutter the history of your project with unneeded commits you may decide to use the amend feature.
Be aware that ammending commits which already visible in other repositories may cause trouble. Once you have pushed a commit to a remote repository or once your local repository was cloned by a somebody else you should be very careful whith amending commits. Simply publishing a second commit which corrects the first one is in the case probably a better solution.
Signing off commits
Git gives you the chance to "sign off" commits. This adds a "Signed off by <Your Name and Email-Adress>" line to the commit message. The purpose of this signed-off-by information is not defined by git but by the team which uses git. Some teams use this information to record who has reviewed a certain commit. Other teams use the signed-off information to record that a commiter agreed to certain standards/rules. E.g. you may have to sign-off your own commits to state that you have read and understood the license-specific rules of your project.