Chassis
A robot without a body is just a bunch of sensors and servos, and maybe a wheel or two. While materials vary in strength, price and workability, a lot of robots have a similar basic structure: There have a few of layers separated by a sheet of material. Strapped to the chassis are the motors and sensors--the tools the robot uses to move and interact with its environment. Atop this layer, there are the chips and motherboards--the brain of the robot. Some designs also have a third layer above that, which may contain other functionality, such as cameras or a payload. Your design should take into account what the robot will do, but new designers should start small.
Sensors and Actuators
In the same way that people have an array of sensors that help them interact with the world, robots need to see, hear or feel their way around the world, as well. There are a variety of sensors that you can use in building a robot, but it will need at a minimum some kind of IR or Sonar device that will give it a picture of where things are in relation to itself.
Actuators act as the robot's muscles, allowing it to actually move in the world and react to its environment. There are a number of ways that a robot can move in the world: Motors move constantly in a direction until told to stop, while servos can rotate to a specific direction with precision.
Software
Now that your robot can see and feel its way around the world, it will need help learning how to do something with that information. Most robots have a small, on-board computer that takes data from the sensors and translates it into motor movement. For your first robot, find a chipset that uses a programming language you are familiar with, or one that is relatively quick to learn--such as C++ or QBASIC.
As you become more experienced, you can start learning about fuzzy logic, which allows the robot to make its own decisions during situations you did not include in your programming. However, this is something incredibly complicated to program, so new programmers won't need to worry about it.
Tips and Warnings
You will want to start making a very simple robot before starting out with the more complicated projects--don't expect to build a replica of Asimo on day one; most beginner's models will have two or three functions at most.
If you are just getting into the hobby, it may be cheaper or simpler to buy a robot kit and start from there (See References).