Ah, professions. Some people love them, some people hate them. Some people tolerate them, since they might give them benefits like an extra 10 points of Stamina or a Uber-Trinket of the Hidden Temple. Professions in LotRO are set up a bit differently than what you're probably used to.
Professions in WoW are fairly straightforward. You can choose any two of eleven possible primary professions, plus you can take all three secondary professions. The primary professions are divided into Gathering Professions (Herbalism, Mining, Skinning) and Production Professions (Alchemy, Inscription, Blacksmithing, Engineering, Jewelcrafting, Leatherworking, Tailoring, Enchanting).
With me so far? I thought so.
In LotRO, you don't quite have the freedom to mix and match professions like you do in WoW, but they make up for that by giving you three professions instead of two. Professions are grouped into clusters called Vocations, and each character chooses one cluster:
Armsman: Prospecting, Weaponsmith, Woodworker
Armorer: Prospecting, Metalsmith, Tailor
Explorer: Prospecting, Forester, Tailor
Historian: Farmer, Scholar, Weaponsmith
Tinker: Prospector, Jeweler, Cook
Woodsman: Farmer, Forester, Woodworker
Yeoman: Farmer, Cook, Tailor
The professions listed in italics are gathering professions, the others are all production professions. Take note that each cluster has one gathering skill and one production skill that work well together: Farming + Cooking, Prospecting + Jewelcrafting, Forestry + Woodworking, and so on. Then they give you one additional skill that doesn't exactly mesh well with the other two, this is to encourage interdependency on others within the community. It's your option if you want to level the third profession up or not. Some players do, some players don't.
Professions in LotRO don't offer passive bonuses or crafter-only items like WoW does, so feel free to choose whatever profession interests you.
Tip: If it's your very first character and you want to make money, try the Explorer vocation. You'll be able to gather ore and wood, both of which sell well to other players. Hides drop off of enemies, and you can use those either to level Tailoring or to sell. If you don't have any interest in crafting at all, but don't mind gathering extra stuff to sell, then Explorer is for you!
Let's go through each profession briefly:
Cook: Produces food. Consumables are very useful in LotRO, either giving you a static buff (+12 Agility) for a certain amount of time or providing a benefit (+100 health and +100 power every 5 seconds) over time. Food is what will help get non-healing classes through tough encounters, and is necessary for instances and raids. Cooks need a Farmer to produce their raw produce.
Jeweller: Makes necklaces, rings, bracelets, etc, similar to the WoW Jewelcrafter. Relies upon a Prospector for ore and ingots.
Metalsmith: Makes heavy armor and shields. Relies upon a Prospector for ore and ingots.
Scholar: Makes potions, buff scrolls, clothing dyes, and housing paints. Scholars are unique in that they collect their own materials: relics found in ruins throughout Middle-Earth. You can buy them on the Auction House, but they tend to be very high-priced. They also use some of the byproducts from Farming to make paints and dyes.
Tailor: LotRO tailors make both light and medium armor. While anyone can collect hides, Tailors rely on a Forester to process/cure the hides before they are able to be used in Tailoring.
Weaponsmith: Makes melee weapons: swords, daggers, clubs, etc. Relies upon a Prospector for ore and ingots.
Woodworker: Makes ranged weapons: bows, staves, and spears. Relies upon a Forester for treated wood.
Prospecting: Mines ore nodes and processes them into ingots for Jewellers, Metalsmiths, and Weaponsmiths.
Forestry: Collects wood for Woodworkers, and cures hides for Tailors.
Farming: This deserves a little extra explanation: While this is technically a gathering profession, it's not gathering in the sense of farming stuff while you're out questing. Farming is similar to a production profession in that you have to actually go to the farm fields and sow your seeds, harvest them, and them process the harvest into usable crops. You can buy your seeds and soils from vendors quite cheaply, but you'll be investing your time into the profession and doing a lot of repeated clicking. If you enjoy taking a break from the action, doing repetitive actions in a farm field while chatting it up in the Shire regional chat, or if you like to multi-task while reading or studying, Farming might be a really good fit for you. If you're more action-oriented, limited on game time, or get bored easily, Farming might be just really frustrating. Cooks use the food that Farmers make, and Scholars are able to use some of the byproducts of Farming for dyes and paints.
Tip #2: If you have two characters that you want to level simultaneously, and you don't mind investing some time into crafting, try the Tinker + Historian vocations. They are very complimentary, as you'll have a Scholar for potions, a Jeweler for jewelry, a Weaponsmith for melee weapons, an ore collector to supply the Jeweler and Weaponsmith, and the Farmer can send raw materials to the Cook.