What type of mount do I need?
There are a few standards when it comes to how a front derailleur is mounted to the bike, the most common one being clamp-on. The clamp-on derailleur is secured to the bike's seat tube using a simple warp around clamp. Front derailleur clamps are designed to work with seat tube diameters of 28.6, 31.8, 34.9 or 38.2mm, however, most clamp-on derailleurs come with shims in order to accommodate multiple seat tube diameters.
Braze-On front derailleurs are used on road bikes and are bolted directly to a tab on the frame. Adapter clamps that allow a braze-on derailleur to be clamp mounted are readily available if needed.
Direct Mount front derailleurs are bolted directly to the frame and are categorized as either Low Direct Mount or High Direct Mount:
Low Direct Mounts come in three varieties: S1, S2 and S3, which are three different standards defining where the mounting holes are located. The best way to figure out which 'S' you need is to either inspect your current front derailleur for a marking or check the frame's documentation. Because of the bolt-on nature of Low Direct Mounts, these derailleurs cannot be adjusted in height or rotation. As a result all Low Direct Mount derailleurs are designed for a specific range of chainrings such as 26-39T, 28-42T or 30-45T. The derailleur must be compatible with the chainrings on the bike. The front derailleur configuration is usually marked directly on the derailleur.
S1 mounts have 42.7mm between the two mounting bolt holes.
S2 mounts have 22.1mm between the two mounting bolt holes, and both holes are on the same plane (ie. flush surface).
S3 mounts also have 22.1mm between the two mounting bolt holes, however the holes are on different planes; the forward bolt has a 5mm offset towards the drive side of the bike.
High Direct Mounts are similar in concept to road bike's Braze-On mounts where these derailleurs are bolted directly to a tab on the frame and allow for height adjustment.
front derailleurs, although somewhat obsolete these days, are mounted to a plate which is held in place by fitting it between the bottom bracket cup and the frame.
Top or down swing, high or low clamp?
Let's first look at what the difference between top and bottom swing front derailleur is. Front derailleurs use specifically designed linkages in order to move in a parallel fashion along the chainrings. Top and bottom swings define the two kinds of linkages that exist.
Bottom swing (aka high clamp) derailleurs are by far the most common types of front derailleurs. The cage on these derailleurs swings below the clamp, hence the term bottom swing. By default, all high direct mount derailleurs are bottom swing type.
On the other hand, the cage of top swing (aka low clamp) derailleurs swings above the clamp, hence the term top swing. By default, all low direct mount derailleurs are top swing type.
Bottom swing front derailleurs are the go-to derailleurs, and the only reason top swing derailleurs are used is when there is no room for a top swing derailleur. The compact design of a top swing derailleur will provide better clearance when needed (for example rear shock clearance), however is not as robust as its bottom swing counterpart.
Do I need top or bottom pull?
When the the shifter is triggered, it is the derailleur cable that pulls on the derailleur linkage to move to chain. If the derailleur cable runs along the top tube of your bike and comes in at the front derailleur from the top, then a top pull front derailleur is needed. On the other hand, if the derailleur cable runs along the down tube of your frame and comes in at the front derailleur from the bottom, then a bottom pull front derailleur is needed. Note that some derailleurs are designed to be dual pull, which mean they will accept either cable coming from the top or bottom.
What is capacity?
Front derailleurs have their limits as to how how big the difference between the largest and smallest chainrings can be. This limit is known as capacity, and is defined as the difference between the largest and smallest chainrings. For example, for a road bike with a 53-39T crankset, the capacity would be 53-39=14T. A suitable front derailleur must then at least be able to handle a 14T capacity, or more.
Double derailleurs are designed for double cranksets, while triple derailleurs are designed for triple cranksets. The difference usually lies in how wide the inner plate of the cage is. Since the difference between largest and smallest chainrings is bigger for triple cranksets than for double cranksets, the inner plate on triple front derailleurs are wider. Note that some front derailleurs are designed to work for both double and triple cranksets.
What is maximum ring?
This one is pretty straight forward and represents the maximum number of teeth a front derailleur can handle.