
Articulating, Semi-rigid, Full-rigid: This is a description of the frame of the crampon. For an articulating crampon, the bar that connects the front and back components will pivot, so the crampon can be fitted to boots that have a flexible sole - that is trekking boots. The sole still needs to be reasonably stiff, ideally with a half-length shank for the crampon to fit securely. The limitation of wearing trekking boots and articulating crampons is that it is not possible to use the front points for steep climbing. Furthermore, trekking boots should not be used with semi or full-rigid crampons as the boots' flex may eventually break the crampon, so to front-point the climber needs a completely stiff soled mountaineering boot. There is no breakage issue in fitting an articulating crampon to a stiff soled boot.

Semi-rigid crampons (left) and fully rigid crampons (right)
However, for a mountaineering boot there is better performance in using more rigid crampons. Semi-rigids have a bar connecting front and back components, which flexes ever so slightly under the weight of the boot while front-pointing, whereas the frame of full-rigid crampons will have no flex at all. Traditionally the thought was the more rigid the crampon the better the crampon was for front-pointing because it provided a more stable base for the boot to stand on. Hence full-rigid classics such as the Footfang were used for the most technical climbing. Sadly full-rigids were also heavier so semi-rigids were preferred by most for long alpine routes.
More recently there has been a shift to using semi-rigids for the most technical climbs (including waterfall and sport climbing) because climbers get a better "feel" for the ice as the crampon moves slightly on the boot when kicked or placed. Hence semi-rigid crampons are now dominating the market for all types of climbing.
12-point, 10-point: Simply the number of points (spikes) on the crampon. 12-points are the standard and 10-points are light-weight crampons used for simple snow climbs or skiing.
Front Points: Horizontal front points are used on general mountaineering crampons (eg. Simond Makalu). For ice climbing it is better to have vertical front points of forged metal that look like a pick of an axe and penetrate ice easier (eg. Simond Vampire). The disadvantage of vertical points on softer snowpacks is they can shear through the ice. This effect is reduced with a horizontal wedge running along the top of the vertical frontpoint.
Horizontal front points on horizontal rails (left) and forged vertical front points on horizontal/vertical rails (right)
Many technical climbers prefer a mono front point (ie. one front point, not two). A mono-point will penetrate ice more easily, can torque into cracks and be swivled on after placing a high kick. A mono-point is a less stable platform to stand on however so duo-points are used for long alpine climbs.
The secondary points (ie. the pair of spikes directly back from the front points) are different for general and technical crampons. For technical crampons they point further forward, acting as a second set of front points for stability.
Rails: The rail is the frame of the crampon. A horizontal frame (ie. horizontal to the ground) has more flex under the weight of the boot - the major benefit being snow won't stick to the base of the crampon. Vertical frames were widely used for technical climbing as they provided a more rigid base for front-pointing but most are reverting back to horizontal rails again - similar to the discussion above on rigid crampons.
Anti-balling plates: Wet snow sticking to the base of crampons is called "balling". It can be reduced with rubber or plastic plates fitted to the crampon. All Simond Vampire and Makalu crampons come with anti-balling plates as standard. Plates that are soft, such as Simond's, are more effective.
Bindings: The binding a climber chooses depends on their boot. All mountaineering boots will have grooves at back and toe for a heel clamp and toe bail, respectively. This is the quickest and most secure form of binding (Simond “speed” binding). Some prefer a heel clamp and toe ring (Simond “mix” binding) if they feel their boots' toe grooves have worn away. This is the least secure system - if the crampon comes loose it can fall off the boot completely as it is not directly tied around the ankle. It is more likely for large sized boots. A slower but better binding is a full strap system (Simond “strap” binding). This is the only option for flexible trekking boots.
Heel clamps should be adjusted so they take effort to snap into place.
Quick Adjust: Simond semi-rigid crampons are quickly fitted to the length of any boot by lifting a lever and sliding the connecting bar to the desired length. The more traditional design is a screw that must be taken out and reset.
Quick adjust system