There are a lot of different ways to build muscle, the most popular one being weight lifting. The advantage of lifting weights is that you can target specific muscles and work just that one muscle group. This is good if you are a body-builder, because they need to have each muscle specifically be as big as possible. But for most people, I don't think this method is the best, for a couple of reasons.
First, lifting weights means that you will get good at...wait for it...lifting weights! The problem with this is that most sports don't have you moving anything but yourself. So for most people, what they want to be good at is moving their own body, and to do that, it is better to do workouts that use your body as the resistance. It may seem limiting, not using any free-weights or machines, but there are so many different ways to lift and push yourself that you can really work every muscle.
Another advantage of these methods is that it improves your balance. For example, if you are doing squats on a machine, you can push a lot of weight, but the machine keeps you from tilting side to side, so it doesn't work those stabilizing muscles. If you do squats without a machine, you will work all of those ancillary muscle groups and as a result your balance and control will improve.
There is also an issue of practicality: if you only know how to work out with machines and weights, and the gym is closed, or you are out of town, or you can't get to that equipment for some other reason, then you have to skip your workout that day. That can be very frustrating, especially if you have specific fitness goals you are working towards and every workout counts. If your workouts use only your body, then you will never be missing what you need. This is very freeing.
So what types of strength training exercises don't use any equipment? Squats, lunges, planks, push ups, tricep dips, ski jumps, and russian twists are a few. Cruising Youtube is a actually a good way to find new workouts, there are so many of them out there, you can constantly be looking for new ones. P90x is a popular system that uses these types of movements, and would work well at getting you strong very fast. A great resource that I found is the Nike Training Club iPhone app; this is a program that has a huge variety of workouts at different difficulty levels, and it has video clips of each exercise to show you exactly how to do it. Plyometrics is another method where you do a series of fast jumps and lunges; they builds strength and agility, and is used by a lot of professional athletes. Yoga is also good: most people think of flexibility when they think of yoga, but the more advanced poses are very strenuous and help develop good muscle tone.