Hammers give the weilder the ability to deliver a sudden shock of force to whatever they hit, which can aid in the loosening of certain parts. The very well-known example of a hammer is a claw hammer, with a wood handle and metal head, but there are many variants, and the claw hammer is rarely used for automotive applications, as it is designed for carpentry use with nails.
Ball Pein/Ball Peen
A ball pein (sometimes spelled "ball peen") is a hammer with a hemispherical head, though it often has a flat head on the other side for increased usability. They used to be used extensively in metalworking, but are still used for that and rounding off edges of fasteners. They are also useful for striking a more precise and small area than a flat-faced hammer face would be able to.
A ball-peen hammer with a flat basic face on the other side of the head is a typical first hammer purchased by an automotive technician, as it is often the most commonly used hammer.
Dead Blow
Dead blow hammers contain loose weight within them so that bounce is greatly reduced upon impact. They are often coated in plastic so that they have a soft hammer face, reducing the potential for damage to components. There is usually a weight listed on the head of the hammer, and the extra weight creates a stronger strike. The core of the hammer is often filled with lead or steel beads.
The bounce other hammers have can cause unwanted damage and redirects energy less effectively. Dead blow hammers are usually coated in orange plastic, but can also be found in green, red, or other less common colors.
Rubber Mallet
Rubber mallets are a basic soft-faced type of mallet that can be used to hit objects with a greatly reduced chance of damaging them. Some designs feature a head made entirely out of rubber, while others (known as soft-face hammers) have a metal head with rubber attachments. The rubber attachment design often has two striking surfaces and are often made of two types of rubber so there are options for firmness. The all-rubber head type of mallet has the advantage of being lighter than most hammers, making it easier to use, especially for extended periods of time. Rubber mallets are a decent substitute for a dead blow hammer when one isn't available.
Brass Mallet/Hammer
Brass hammers can be used for hitting softer metals, and the head of the hammer may deform instead of the surface being hit. A brass hammer also has a reduced risk of generating sparks when striking, but won't prevent them entirely.
With use, brass hammers tend to mushroom out and for safety, the spread out head needs to be ground off so that the pieces don't chip off and injure the user or someone nearby.
Sledgehammers
Sledgehammers are extra heavy hammers designed for maximum striking force above all other concerns. Not all of them have a long handle, but it gives the hammer extra swing potential for a harder strike. The long handle versions typically aren't used by automotive techs, unless whatever is being struck is not going to be reused.
Sledgehammer weight varies, but they are typically the heaviest form of hammer used by humans, rather than machines.