We aren't going to define all the terminology used in mountaineering. This is a list of some of the more confusing terms for those starting out in mountaineering.

Climbing

crux: The hardest part of the climb. Stick around long enough and someone will tell you that the crux of the climb is getting up for an alpine start.

alpine start: Getting up in the early morning for a climb. This is often as early as midnight. The point is that you get better snow conditions in the early morning.

freeze: A freeze refers to the snow becoming harder overnight and thus allowing easier progress to be made (e.g., did you get a good freeze on your climb?).

freeclimb: The most misused word in climbing. It does not mean to climb unroped (which is actually called soloing (although read the solo entry for further clarification)). Free climbing means you are not aid climbing. This means that all of the climbing you have ever done is likely to have been freeclimbing.

aid climbing: Aid climbing involves pulling yourself up on protection (pitons, wires, ice screws, etc.) that you have placed rather than climbing up on handholds and footholds. Climbers can aid climb for pitch after pitch, in which case they will be using ladders made of sling to stand in and other specialised equipment.

big wall: Usually a big wall is a mountain face that is typically climbed with at least some aid over a number of days.

alpine style / siege style: Alpine style refers to climbing from the base of a mountain to the top directly, whereas siege climbing refers to setting up camps part way up the mountain and stocking them with supplies for a later summit attempt. Alpine-style climbing may take more than one day and can involve camps, but the climbers only ever continue upward. To climb without camping is to climb in a single push. All this terminology is usually only used for high-altitude climbing (e.g., Himalayas, Alaska, Karakorum).

technical: A technical climb is one that involves climbing rather than walking (e.g., the top 100m of the climb was quite technical). A technical axe is a short axe (usually 50cm in length) used for steep climbing.

rap/abseil: Rapelling/rapping just means to abseil (e.g., we rapped the descent route).

mixed (dry-tooling): Mixed climbing can be 1) climbing on a mix of ice, snow, rock and frozen turf or 2) climbing rock while wearing crampons. The idea is that you aren't simply climbing on snow or ice with your axe and crampons or climbing rock with your hands and boots. Dry tooling refers to using your axes and crampons while climbing rock.

pitch: This word has a number of meanings. It can refer to a ropelength of climbing (e.g., the climb was 15 pitches in length; we had trouble on the third pitch of the climb). It can also refer to using a rope and other gear to prevent a fall in a particular manner. In this latter case, only one climber is climbing at a time while the other is belaying (e.g., we decided to start pitching since the ground was getting steeper).

solo: Soloing has two quite different meanings: 1) climbing alone (with or without using a rope) and 2) climbing without using a rope (but not necessarily climbing alone). If you are climbing alone but using a rope (quite a complicated process), then you are rope-soloing. If you are climbing alone and not using a rope, then you are free-soloing.

lead/second: Leading and seconding refers to pitch climbing. The leader is the climber climbing above the belay, placing protection as they go. They face the challenge that if they fall, they will fall a certain distance before the rope catches them. The seconder has it easier in that they are being belayed on a tight rope by the leader higher up (e.g., the climb is a scary lead, but it is straight-forward to second).

belay: This has two meanings. First, it is the act of paying out and locking off the rope going to the climber (e.g., my partner belayed me across the snow slope). Second, it is the protection securing the belayer to the mountain (e.g., the belay comprised a piton and snowstake; we found a good (site for a) belay 30m higher up the ridge). In this latter case, we can refer to a belay as an anchor (e.g., I set up an anchor to belay my partner across the slope). Whatever we abseil from is referred to as an anchor, but not as a belay (e.g., we abseiled off an anchor comprising a sling around a rock bollard).

runner: Runners are the protection placed between belays to reduce the distance fallen by the leader (e.g., I placed a runner 10m above the belay). The same word can be used to refer to a sling running from the protection to the rope (e.g., I used a long runner so that the rope ran freely past the rocks).

simulclimb: Both climbers climb at the same time with the rope extended between them with the leader placing protection as runners but not making belays. This is also referred to as climbing on runners.

pro(tection)/gear: Anything used to anchor you or the rope to the mountain, such as snowstakes, ice screws and pitons (e.g., what gear did you place?; the protection wasn't very good).

front pointing: Climbing on the front points of your crampons.

Mountains

(berg)schrund: The crevasse that separates a glacier from the mountain slopes above. Schrunds often pose an obstacle to cross at the start of a climb.

icefall: Part of a glacier that is in a real mess. If a glacier can be thought of as a frozen river, then an icefall is the rapids.

serac: A tower of (often unstable) ice in an icefall.

cut up: A glacier is cut up if there are many open crevasses through which navigation is difficult.

ice cliff: A wall of ice that can be severely overhung and unstable and is formed by a glacier high on a mountain side flowing over a rock cliff. Ice cliffs are dangerous things that you don't want to stand under.

couloir: A narrow gully on a mountain face.

col: a pass (saddle) on a ridge, usually not prominent enough or low enough to be referred to as a pass.

Equipment

pins/pegs/pitons: These are all names for the same type of protection.

wires/nuts/jamnuts/stoppers: These are all names for the same type of protection.

half rope/twin rope/full rope: All these types of ropes can be of any length. They differ in their diameter, or more technically, their stretch since thinner ropes stretch more.

tool: Another name for an ice axe or hammer. More specifically, it refers to a technical climbing axe or hammer rather than a walking axe.

shell: A rainjacket (i.e., something completely waterproof). Shellwear refers to waterproof jackets and pants. A shell is sometimes referred to as a hardshell to differentiate it from a softshell.

softshell: A garment that will keep off rain to some extent but is not completely waterproof.