Quite an old article on mssqlcity attempts to explain the types of locks:
Shared locks are used for operations that do not change or update data, such as a SELECT statement.
Update locks are used when SQL Server intends to modify a page, and later promotes the update page lock to an exclusive page lock before actually making the changes.
Exclusive locks are used for the data modification operations, such as UPDATE, INSERT, or DELETE.
What it doesn't discuss are Intent (which basically is a modifier for these lock types). Intent (Shared/Exclusive) locks are locks held at a higher level than the real lock. So, for instance, if your transaction has an X lock on a row, it will also have an IX lock at the table level (which stops other transactions from attempting to obtain an incompatible lock at a higher level on the table (e.g. a schema modification lock) until your transaction completes or rolls back).
The concept of "sharing" a lock is quite straightforward - multiple transactions can have a Shared lock for the same resource, whereas only a single transaction may have an Exclusive lock, and an Exclusive lock precludes any transaction from obtaining or holding a Shared lock.