Off-camera flash is a great way to augment your existing photographs. There are so many times when existing light just doesn't give you the result you desire, and that flash could be a solution to creating the image you have in your mind. When you first start, however, the options can be quite overwhelming and it can be difficult to know exactly what you'll need. Let's look into a simple but versatile kit that will allow you to stay mobile and work in many different situations.
You'll need a flash. But what kind? How much should you spend? Since we'll be dealing with off-camera flash here, it doesn't really matter which brand you buy. I'm going to recommend you look into Nikon's previous generation of high-end flashes: the SB600 and SB800. These can be had for around $80 or $100 respectively second hand and will do everything you need and more. Of course, you could look into off brand flashes like those made by Yongnuo or LumoPro, but so far, in my experience, they just don't last. I have dragged my SB800s through 15 countries, and all conditions from ice to desert, and have never had a hiccup. These are exceptionally well built, light weight, powerful, and relatively cheap. Start with one of these and you won't be looking for another flash for a while. If you're made of money, you could of course invest in your brand's flagship speedlight like these from Canon and Nikon.
The major benefits of the SB800 over the SB600 are an extra stop of light, a better interface, and a PC-sync port. The extra light is always nice to have, especially if you're working outdoors. The interface? It's just more robust. You can pick and choose all your options from a menu that pops up on the large LCD, giving you quick access to everything the flash has to offer. The PC-sync port gives you wont more way to fire the flash in a pinch. If you're a Nikon user, the SB800 is a no brainer. It can command other flashes in Nikon's CLS system, which can be great if you don't want to work with radio slaves.
Once you get that flash off your camera, you're going to want a way to trigger it. You could do this via a cable, optical sync, or via your camera's built-in control system (brand dependent). However, in all honesty, you're going to want some radio triggers. They just work. Again, plenty of super cheap options exist, and lots of very expensive ones. My recommendation is Impact PowerSync16, which are made by SMDV in Korea. I've been using the series three triggers for five years now, in all the same conditions as those SB800s. Not even one misfire. Not one. These are excellent value for money. A pair can be had for around $160. The battery in the trigger lasts forever, as do the AAs in the receiver. They're basic, in that they don't offer TTL or any other fancy features, but they will get you through every single shoot without a hitch. One of the other benefits is that unlike other options such as the Pocketwizard Plus X line, these have a hotshoe, meaning that you won't need any extra cables or a PC sync port on your flash.
So it's off your camera and on a trigger now. What next? You'll want a place to put it. Grab yourself a light stand. Personally, I use the Manfrotto 5001s. They fold up small, and they're super light. You could use something cheaper, like these ones from Impact. These are a great option if you're out of the wind. However, I like the 5001s because they go flat to the ground, so I can put rocks, bricks, or even my camera bag on them for a little extra stability outdoors. However, I shoot mostly people in the 5 to 6 foot range. If I were shooting taller folks, I'd be looking into something with a little more height.
You'll then need a way to mount your flash and modifier to that stand. I recommend getting the heavy-duty Manfrotto Swivel. Again, there are cheaper options, but this will last. I've been using the same ones for nine years now, and haven't had a single issue with them.
Right in the beginning, I'd suggest a medium sized umbrella. Impact and Westcott both have some great, and really cheap, options. You'll just want something to soften up your light. Of course, you could go DIY and use a bed sheet or some tissue paper, but with the price of umbrellas, I'd recommend getting one for the ease of use. Something around 32 inches is a great size to start with. It will allow you to get soft light on people at a reasonable distance.
Once you're done with this basic kit, there are a couple of other things you might want to add before you start looking at bigger lights or expensive modifiers. These are gels and batteries. Gels are little pieces of colored material you can place in front of your flash to either match it to the color of existing light (like an orange gel for tungsten) or for creative effect (like a purple gel for mood). These will give you some fun creative options once you've mastered the ins and outs of basic flash use. I'd also recommend a good set of NiMH rechargeable batteries like Eneloop Pros. Not only will these give you more juice and save you throwing out so many batteries, they will also cut down your recycle time between flashes.