All weapons capable of firing multiple rounds through a single barrel without reloading have a magazine of some sort. This obviously means that break-action rifles and double barrelled shotguns (as well as some elaborate pepperbox derringers) are not magazine fed.
Internal vs Integral Magazines
Almost all modern rifles and pistols use detachable magazines of some sort. Early rifles tended to use internal magazines, loaded via stripper clips or en bloc clips. The M1 Garand is probably the best-known en-bloc-clip-fed rifle in Western history.
Examples of Internal Magazines
Lever-action guns typically have a tubular magazine located beneath the barrel.
M1 Garands have magazines which require an en bloc clip in order to feed rounds. The en bloc clip is not readily removable, which is why it is not considered a magazine.
Mosin-Nagant rifles all feature 5-round internal magazines.
Pump-action shotguns generally use internal tubular magazines.
Revolvers have cylinders, which are a form of magazine. They are usually not designed to be removed for quick-change, and are reloaded instead with moon clips or speed loaders.
SKS Rifles, other than a few after-market models designated as "SKS-D", "SKS-M", or "SKS Sporter", have fixed magazines of a similar design to Mosin-Nagant magazines. They are meant to be loaded via stripper clips.
Various old Mossberg 22LR rifles have tubular magazines which feed from the rear of the stock and proceed upwards to the action. They use a spring-loaded rod, which serves as an endcap to the tube, to push rounds from the rear of the rifle and up into the action.
Examples of Detachable Magazines
Calico rifles use a design called a "helical magazine". This is a magazine design in which the rounds are stored in a "corkscrew" pattern, to increase the number of rounds which can be stored in one place. Helical magazines tend to lie along the top of the weapon, with the feed lips on the bottom of the magazine.
M1919's, and other belt-fed rifles, are considered to be magazine-fed. The belt constitutes an "ammunition feeding device". In states with magazine capacity laws, this can become an interesting conundrum.
PS90s (and their military P90 brothers) use non-traditional box magazines, in which the rounds are stored pointing 90 degrees away from the direction of the barrel and rotate 90 degrees when reaching the feed lips.
STANAG magazines are the ubiquitous AR-15 magazines. These are frequently made from aluminum, although several companies now make stainless steel variants.
Thompsons, available in SMG form and rifle form, use either normal box magazines or traditional drums. In a drum, the rounds are stored in a "snail shell" type spiral.
Standards in Magazines
The most prolific standardized magazines in the world, are the AK-47 family magazines and the STANAG NATO magazine. If you think you've never heard of a STANAG magazine, it's only because you know it as an "AR-15 mag" or "M-16/M4 mag". I am not aware of any non-AK rifles which accept AK magazines, but would welcome information on any.
Rifles which accept STANAG magazines
For no reason other than to stretch my memory, I shall endeavor to list a number of rifles which accept STANAG mags.
AR-15 family (too many members to begin listing)
Beretta AR-70/90 (the first-generation AR-70/223 used a proprietary magazine)
Bushmaster ACR (formerly known as the Magpul Masada)
Galil 223, with adaptor (non-adapted rifles use proprietary, more AK-like Galil magazines)