You've made a good decision to ask about these different systems.
Whether the weapon uses a hammer or not, and whether the hammer is exposed or enclosed, may not be very important. I'm not aware of any handguns at all that use an internal hammer, although I don't doubt there may be many such designs during the 20th century.
Handguns with hammers generally have a relatively small, short firing pin loaded with a spring that holds the pin rearward, and keeps the back end of the firing pin exposed and 'proud'--meaning the back of the firing pin is held up higher than the surrounding surface so it can be struck by the hammer.
When the trigger is pressed and the hammer falls, the hammer strikes the pin, thrusting it forward against the restraining pressure of the firing pin spring so it strikes the primer of the cartridge.
Generally, there are a variety of safety mechanisms that have been used in a variety of combinations--blocks that prevent the hammer from falling, blocks that prevent the firing pin from moving forward unless the trigger is pressed, and blocks that prevent the trigger from being pressed at all.
Often, these mechanisms allow the gun to be carried with the hammer cocked (as in the case of the single-action M1911 design). To fire the gun, any safeties must be released, and pressing the trigger releases the hammer. In such guns, the trigger has just one function: releasing the hammer.
In other designs, the gun is carried with the hammer lowered using what is called a 'decocker'--a lever the operator uses to safely lower the hammer. These designs are generally used in what is called a 'double action, single action' (or DA/SA) pistol, where the gun is initially fired by a trigger pull that first cocks then releases the hammer. During this first trigger press, the trigger serves two distinct functions (hence, 'double action'). It cocks the weapon, AND then releases the hammer. After this initial firing, normal cycling of the slide in firing the gun then cocks the hammer, and the trigger then serves just one function, to release the hammer ('single action). Most Sig-Sauer pistols use this design, as do many other designs (e.g., CZ).
In the designs mentioned above, the firing pin is a cylindrical rod, held to the rear by a light spring that simply ensures the back of the pin is exposed and can be hit by the hammer, and that ensures the front of the pin is NOT exposed at the breech of the weapon to ensure smooth pick-up and feeding of the cartridges.
Where you see no hammer on a handgun, the firing pin is of a different design completely, and is referred to as a striker, as already mentioned. Now, the firing pin spring (or striker spring) serves a completely different function, and is used to thrust the strker forward into the cartridge primer. In these systems, the gun is cocked by the forward travel of the slide in which the striker rides. As the slide moves forward into battery (its final forward position), a fixed 'leg' on the striker is snagged on the sear, which is mounted to the frame. The sear prevents the striker from continuing forward with the slide, but the slide continues forward. This action compresses the striker spring and, depending on the particular design, either fully or partially cocks the weapon. Cocking is complete in some guns such as the Smith & Wesson M&P design, or only partially cocked as in the case of the Glock design.
Now, pressing the trigger either a) lowers the sear down, releasing the striker (in the case of a fully-cocked striker), or b) further compresses the striker spring to fully cock the striker--further trigger motion then releases the fully-cocked striker. In both cases then, the striker flies forward due the compressed striker spring.
Typical striker, striker spring and sleeve assembly. The arm that catches on (and that is released by) the sear happens to be pointed up in this photo. In the gun, this arm points down, since of the course the sear it engages is in the frame of the weapon:
Because there is no hammer to carefully lower to decock these guns and no hammer to cock manually, these designs are generally always cocked--either fully or partially. They can be decocked only by pressing the trigger and firing them on an empty chamber. But, they all rely on a variety of internal safeties, and some provide additional external (manual) safeties as well. These safeties, as in the case of many external hammer guns, physically block the travel of the striker with a steel block. This striker block is spring loaded so as to always block any possible forward motion of the striker.
To allow firing the weapon, the striker block has to be moved up and out of the way of the striker, and the only way this is possible is through pressing the trigger--the trigger MUST be pressed to slide the striker block out from in front of the striker, and therefore in all striker-fired weapons I'm aware of, the guns cannot fire under any circumstances unless the trigger is pressed.
The addition of an external ('thumb') safety can be used to prevent the trigger from being moved at all.
Note that a 'striker' isn't different in any basic way than the firing pin in the bolt of a 'bolt action' rifle. It is nearly identical in design to that mechanism, used now for well over a century.
These are the high-level basics. More details would provide a more accurate and complete description, but I think you've got what you need now to understand that, whether or not you see a hammer on the gun can only tell you so much about how the gun functions and what that impacts as far as operation and safety.
You can now google topics such as 'striker vs hammer', 'SA/DA guns', 'handgun design safety' and so on. There are many, many articles on these topics. Do NOT buy a handgun based on whether it has a hammer or not--it simply doesn't tell you very much.