if you look at the Online Package Manager there is a GifAnim package you can install..
I tried to install "Online Package Manager" package and found this error "Cannot find lazversion used by BGRAFreeType of package BGRABitmapPack"
There's something wrong here... What is your Lazarus version? Throughout the entire v2.0.x series of Lazarus Online Package Manager is installed by default, and if you really need to install it it does not depend on BGRABitmapPack.
If you have a version older than Laz 2.x you should take the chance to upgrade; this way you will get a new IDE and get rid of this misconfiguration.
If you already have 2.0.x I'd recommend that you reset your user settings: Open "View" > "IDE Internals" > "About IDE" and find the line "Primary Config Directory" - it tells you the folder in which the user settings of your installation are stored. Close Lazarus, go to this primary config folder and rename it (to have a backup when something goes wrong). Open Lazarus again, it will complain about packages not found, ignore these messages. Go to "Tools" > "Configure Build Lazarus" and check "Clean all" and "Switch after building to automatically". Then "Build". After a while Lazarus restarts and only contains the default packages. Go to "Package" > "Online package manager", check the "GifAnim" package and install it. It must work - I did it many, many times (ok, not with GifAnim...)
Be warned: this way you lose all your own configuration settings which you must reselect. And you lose all your manually installed third-party components which you must reinstall again.