An amendment is a document whereby the lessor and lessee agree to change a lease. If the original lease contained an error, or if new terms were later negotiated, then an amendment is needed. Oftentimes, the amendment will be dated effective as of the date of the original lease, in which case, you can treat it as if its terms were in place back when the original lease was signed.
Unsurprisingly, an extension is just what it sounds like. It is a document by which the parties to a lease agree to extend the primary term for a certain number of years. It allows a lease to continue in effect, without the need for production, for a period of time longer than what was originally agreed to. A new bonus payment to the lessor generally accompanies an extension.
A release releases the parties from their obligations under a lease. A partial release releases them as it relates only to specific acreage and/or depths. In the case of a partial release, the lease would continue in effect except for whatever portion was released.