Mountaineering - a very general term for moving through, and over, the mountains on foot. May include some hiking, some moving over snow, ice and/or rock, glacier travel or all of those. May include technical climbing, or just hiking in the mountains. Does not include rockclimbing at the crag, bouldering or crag-style WI ice climbing.
A subset of Mountaineering is ...
Alpine Climbing - climbing peaks in an alpine environment, usually of a more technical nature than just hiking up mountains, though hiking may be involved in approaching the more technical climbing. May involve rockclimbing, ice climbing, snow climbing and/or glacier travel, maybe all in the same morning. An alpine climb may often involve snow or ice climbing, and glacier travel, but in some cases can be a predominately rock climb. So, longer, higher and colder than just rockclimbing at the crag. More technical than just hiking up a hill. Done on Lower 48 peaks, Alaskan peaks, Himalaya, European Alps, New Zealand, Patagonia etc - anywhere there are relatively high peaks that require long climbing sessions to get up them.
Those who undertake alpine climbing on bigger mountains may do so in ...
Alpine-Style - a method of climbing mountains. From the bottom to the top in one go, maybe stopping to camp along the way, but no camps are left on the mountain, no going up and down to carry loads, no fixed rope, no Sherpas or high altitude porters (ie. no porters above base camp). Ideally, no prior inspection or climbing on the route. Used on Andean, Alaskan, Himalayan type peaks, though the origin of the term came from trying to climb in a style similar to that used in the Alps - the European Alps - as in using no camps on the mountain, just up and down in one go (often one day).* It is often misused in various ways, such as "we climbed alpine-style from the top camp" - which is an inherently impossible thing to do.
Alpine-Style is usually considered a stylistic, environmental and sporting improvement on ...
Siege Style - gradually placing higher camps, usually with fixed ropes between them to enable shuttling up and down, use of high altitude porters / Sherpas. Nowadays usually only used on bigger Himalayan / Karakoram peaks, though it was the method used for climbing many big peaks earlier last century.**
So, 'alpine climbing' and 'alpine-style' climbing are two very different things. 'Expedition style' is an inaccurate and misleading term because a mountaineering expedition to a peak can climb that peak in alpine-style, as many do.
* A further variant to alpine-style is 'single push' climbing, which in current use really means 'non-stop single push' where the climbers go for a very long time to complete the route and descent in a continuous session, maybe more than 24hrs, not even camping or bivying, primarily to avoid carrying heavy loads and to make use of short weather windows. This is a more accurate translation of the true style of climbing in the Alps to the Greater Ranges than 'alpine-style', as camps are rarely used in the European Alps. This method is still quite rare, but was made famous in the 80s and 90s by Euro alpinists like Erhard Loretan, Jean Troillet, Pierre Beghin, Voytek Kurtyka and Benoit Chamoux. It is from this era and activity that the term 'fast and light' (actually léger express) was coined, though that is now applied to other less impressive ascents.
** A method that is in-between alpine-style and siege-style is capsule-style. This is moving up the mountain in one go, but fixing ropes just between two camps and shuttling back and forth between just those two - never back to base camp, always staying on the route. The two camps and the fixed lines between them form the 'capsule' and that capsule gradually moves up the mountain, unattached to base camp. Once a new camp is established above the top one, the bottom one is dismantled and the ropes above it used higher up to place another camp. This is not used often, mostly on very difficult new climbs on big steep peaks in the Greater Ranges. eg. Changabang, Trango, Thalay Sagar etc.