The latch mechanism is operated by a standard door knob.
The latch mechanism is operated by a lever.
The latch mechanism is operated by a thumb latch.
Mortise locks are usually used for commercial applications. The latch and the locking mechanism are self-contained within the door.
Single cylinder deadbolts are standard for most applications. However, double sided deadbolts are available where the lock is keyed on both sides of the door.
Most locksets of any type can be rekeyed to your needs and are available in different functions. Different functions available include storeroom (locked at all times), classroom (can be left locked or unlocked via a key), passage (unlocked at all times), privacy (locked on button on the inside, commonly used for bathrooms), and many more.
All locksets are available in many designs and finishes.