You may find these things in your git repository (.git directory for a repository associated with your working tree, or 'project'.git directory for a public bare repository).
Object store associated with this repository. Usually an object store is self sufficient (i.e. all the objects that are referred to by an object found in it are also found in it), but there are couple of ways to violate it.
You could populate the repository by running a commit walker without -a option. Depending on which options are given, you could have only commit objects without associated blobs and trees this way, for example. A repository with this kind of incomplete object store is not suitable to be published to the outside world but sometimes useful for private repository.
You also could have an incomplete but locally usable repository by cloning shallowly. See linkgit:git-clone.
You can be using objects/info/alternates mechanism, or $GIT_ALTERNATE_OBJECT_DIRECTORIES mechanism to borrow objects from other object stores. A repository with this kind of incomplete object store is not suitable to be published for use with dumb transports but otherwise is OK as long as objects/info/alternates points at the right object stores it borrows from.
Traditionally, each object is stored in its own file. They are split into 256 subdirectories using the first two letters from its object name to keep the number of directory entries objects directory itself needs to hold. Objects found here are often called unpacked (or loose) objects.
Packs (files that store many object in compressed form, along with index files to allow them to be randomly accessed) are found in this directory.
Additional information about the object store is recorded in this directory.
This file is to help dumb transports discover what packs are available in this object store. Whenever a pack is added or removed, git update-server-info should be run to keep this file up-to-date if the repository is published for dumb transports. git repack does this by default.
This file records paths to alternate object stores that this object store borrows objects from, one pathname per line. Note that not only native Git tools use it locally, but the HTTP fetcher also tries to use it remotely; this will usually work if you have relative paths (relative to the object database, not to the repository!) in your alternates file, but it will not work if you use absolute paths unless the absolute path in filesystem and web URL is the same. See also objects/info/http-alternates.
This file records URLs to alternate object stores that this object store borrows objects from, to be used when the repository is fetched over HTTP.
References are stored in subdirectories of this directory. The git prune command knows to keep objects reachable from refs found in this directory and its subdirectories.
records tip-of-the-tree commit objects of branch name
records any object name (not necessarily a commit object, or a tag object that points at a commit object).
records tip-of-the-tree commit objects of branches copied from a remote repository.
records the same information as refs/heads/, refs/tags/, and friends record in a more efficient way. See linkgit:git-pack-refs.
A symref (see glossary) to the refs/heads/ namespace describing the currently active branch. It does not mean much if the repository is not associated with any working tree (i.e. a bare repository), but a valid git repository must have the HEAD file; some porcelains may use it to guess the designated "default" branch of the repository (usually master). It is legal if the named branch name does not (yet) exist. In some legacy setups, it is a symbolic link instead of a symref that points at the current branch.
HEAD can also record a specific commit directly, instead of being a symref to point at the current branch. Such a state is often called detached HEAD, and almost all commands work identically as normal. See linkgit:git-checkout for details.
A slightly deprecated way to store shorthands to be used to specify URL to git fetch, git pull and git push commands is to store a file in branches/'name' and give name to these commands in place of repository argument.
Hooks are customization scripts used by various git commands. A handful of sample hooks are installed when git init is run, but all of them are disabled by default. To enable, they need to be made executable. Read hooks for more details about each hook.
The current index file for the repository. It is usually not found in a bare repository.
Additional information about the repository is recorded in this directory.
This file helps dumb transports discover what refs are available in this repository. If the repository is published for dumb transports, this file should be regenerated by git update-server-info every time a tag or branch is created or modified. This is normally done from the hooks/update hook, which is run by the git-receive-pack command when you git push into the repository.
This file records fake commit ancestry information, to pretend the set of parents a commit has is different from how the commit was actually created. One record per line describes a commit and its fake parents by listing their 40-byte hexadecimal object names separated by a space and terminated by a newline.
This file, by convention among Porcelains, stores the exclude pattern list. .gitignore is the per-directory ignore file. git status, git add, git rm and git clean look at it but the core git commands do not look at it. See also: linkgit:gitignore.
Stores shorthands to be used to give URL and default refnames to interact with remote repository to git fetch, git pull and git push commands.
Records of changes made to refs are stored in this directory. See the documentation on git-update-ref for more information.
Records all changes made to the branch tip named name.
Records all changes made to the tag named name.
This is similar to info/grafts but is internally used and maintained by shallow clone mechanism. See --depth option to linkgit:git-clone and linkgit:git-fetch.