If you aren't entirely sure what a profession is, you'll figure it out as we describe them. You can first pick up a profession at level 5, and trainers can be usually be found both in the starting areas and in major cities. (Trainers for skill levels over 300 are only found in Outland, however.)
Professions in World of Warcraft are split into two basic types: primary and secondary. You may only chose two primary professions, while you can have as many secondary professions as you'd like. (Though, at present, there are only three secondary professions.) We'll start out by discussing the secondary professions, since they're applicable to every player.
First Aid: Allows you to create bandages which you can use to heal yourself.
The bandages first aid allows you to craft can be used in and out of combat, though their healing abilities are interrupted by damage. However, they're great to quickly heal yourself up during downtime or give yourself a tiny boost in combat (before they're interrupted). Plus: cheaper than potions. Since it doesn't count against the number of primary professions you can have, there's no reason not to pick up first aid. No reason at all! Even if you're a healing class, you ought to train up first aid -- there are going to be times when you'll be out of mana and glad you had it. So if you don't have first aid yet -- go out and train it. Right now! Hurry up!
Fishing: Allows you to catch fish (and other things) with a fishing pole.
In my humble opinion, fishing is the most boring, awful, and tedious profession in the entire game. The fishing process involves casting your line and then staring at your bobber, waiting for it to go under water, at which point you can click on it for a chance at getting a fish. It doesn't provide a lot in the way of fun or exciting gameplay, but let me tell you, the fish you can catch are the absolute best way to train cooking quickly. Also, there are certain types of fish used in cooking and alchemy that will always sell well on the auction house.
So, my advice -- if you have the patience for fishing (I'm afraid I don't), train it. It'll help your cooking and what you don't use, you can probably sell.
Cooking: Allows you to create food that will heal and buff anyone who eats it.
The food you can make with cooking provides useful buffs for all classes. (Of course, when eaten, food will also heal you, but it's the buffs that are the really cool part of the food items you can get via cooking.) Extra stamina and spirit is a common buff, but at higher levels you'll also come across food that improves your agility, mana regeneration, health regeneration, attack power, healing, and spell damage. Not bad for a secondary profession.
It doesn't train up as quickly as first aid, but it's universally useful and it doesn't take up a profession slot -- so I recommend it for everyone.
Gathering Professions: These professions focus on gathering items that are used by other professions. They're often paired with a crafting profession that uses the gathered materials, but you may elect to train two gathering professions instead. When you're selling the materials you gather instead of using them, you're making a lot more money than you would by using up gathered materials to craft!
Herbalism: Allows you to collect herbs.
Herbs are primarily used in alchemy, but a few herbs are also needed by some recipes in all crafting professions. Certainly grab this if you're planning on picking up alchemy, but if you're looking to grab two gathering professions as moneymakers, this wouldn't be my first choice.
Skinning: Allows you to skin animals for leather and hides.
Leather and hides are both used by the leatherworking profession, so if you're interested in that profession this is a good companion to it. However, I also advise it as a moneymaking profession, because, unlike the other two gathering professions, it doesn't have require a tracking ability. (Both herbalism and mining have special tracking abilities that allow you to see herbs and mining nodes on your mini-map. But since you can only use one tracking ability at a time, skinning is a good companion to any other gathering skill you might chose.)
Mining: Allows you to mine ore.
The ore and gems you mine can be used by Blacksmithing, Engineering, and Jewelcrafting. If you're interested in any of those professions, mining is a good choice because mining your own ore is going to be cheaper than buying it. And because so many different professions can make use of mined ore, it's an excellent choice for making money as well.
Crafting Professions: These professions don't tend to be money-makers. In fact, you'll often find yourself losing money by crafting, since many of the materials you'll use up making your crafted items will sell for more than the items themselves! However, these professions can be useful for providing yourself with items and gear -- and there is the occasional highly desirable crafted item that's worth selling.
Alchemy: Allows you to create a variety of potions and elixirs, as well as transmute certain items.
The potions alchemy gives you access to are always useful for buffing yourself or healing yourself. However, a recent change to potions allows you only to have two types of potions active on you at any given time, meaning you can no longer stack potion buffs indefinately. However, it's still a very handy profession to have, both to help you as you level up and to help you out end-game.
Blacksmithing: Allows you to craft heavy armor (mail and plate) as well as a variety of weapons.
At low levels, blacksmithing will provide a leveling Paladin or Warrior with decent gear. At later levels, blacksmiths must chose their path -- armorsmith or weaponsmith. While there are some worthwhile weapons available, it seems as though armorsmith is the best choice unless there's a weaponsmith-specific weapon you can't live without, as it provides you with an epic quality set of armor in the end. And don't forget -- you can make skeleton keys to open any lock!
Leatherworking: Allows you to craft medium armor (leather and mail) as well as a variety of armor kits.
At low levels, leatherworking will provide a leveling Rogue, Druid, Hunter, or Shaman with useful gear. Later in the game, you'll probably find a lot of the items you can make will be replaced by instance drops or PvP rewards, you'll still find the profession handy for resistance gear and the armor kits you can add to existing items.
Tailoring: Allows you to craft cloth armor and bags.
For cloth-wearers, tailoring provides useful armor both when leveling and when at max level -- as well as spell-thread, which can add to higher level gear to improve it. And the usefulness of crafting bags cannot be overlooked, especially at low levels when bags are rare and expensive.
Engineering: Allows you to create a variety of trinkets and gadgets that are situationally useful.
Engineering is widely considered to be one of the games' least-loved professions, but it is of situational use to anyone interested in its abilities. Engineerers can craft a variety of situationally useful toys and trinkets that are just as likely to harm you as to help you. Paladins find engineering appealing because it gives them access to ranged attacks that they wouldn't otherwise have while PvPers may find that it gives them a handy bag of tricks to draw on in difficult situations.
Jewelcrafting: Allows you to craft a variety of jewelry and (at later levels) cut gems for socketing.
Jewelcrafting is particularly handy because it gives you access to rings, necklaces, and trinkets at a lower level than you'd otherwise have access to them. And at higher levels you can cut gems for socketing, and cut gems are always in demand.
Enchanting: Allows you to enchant gear with additional abilities.
Yes, there is only one "service" profession, and we call it a service profession because the items you create with it aren't tangible objects that can be bought or sold. Instead, the enchantments you create are applied to already existing objects. (This can be something of a pain, because while other professions can buy or sell things on the auction house, you can only give out your enchantments through the trade window.)
While enchanting is incredibly useful at both lower and higher levels, it's also both expensive and difficult to train. The materials you need to enchant something are only acquired by disenchanting magical items, so you'll find yourself disenchanting useful items instead of selling them in order to have raw materials to work with. Also, the enchants everyone wants to buy are often the rare and hard to find ones -- so to make this profession pay off in the end, you'll have to be willing to spend some time to locate the most wanted enchanting patterns.