Robotic sensors are a central part of a robot design. Without sensors, there is no interaction. And without interaction, your robot would just be another machine.
Robots really are machines that can processed information on their own to decide which action they do next. Even if you don't work on mobile robots, you will need sensors to build a robot that can do something on its own.
Now I'm going to give you a list of common robotic sensors and ways to use them in your project.
There are a few sensors that will generally represent the back bone of your robot. For example, most professional robots use encoders to know their position with a very high accuracy.
Encoders are look like a gear, with one part attached to the base, and the other to the wheel or arm. So they will measure the real angular position of the wheel/arm. It's extremely useful since you can never be 100% sure that your motor will do exactly what you because it could be blocked, lose power, and many other perturbations.
While you may not use it on your small robots, a robot like Nao have encoders for all its articulations.
Another sensor that you will find almost everywhere is the bumper or switch. These are so simple, and so extremely useful. You can use them like Roomba to detect walls. You can use them as switches to turn your robot on/off, change mode, or whatever you need. You can also use them to interact: Nao has more than 10 touch sensors or bumpers, and many applications use them to stop, pause, or comfirm actions, or even replace the voice control when it just doesn't understand.
If you build your own robot, for example an Arduino robot, you may need some of the sensors in this section.
Photocells are super useful. You can build a line follower with these, or a coakroach robot that follows or avoid light. You can check if it's the day, the night, or the night with the ceiling light on. You can make a robot chicken that wakes you up when the sun rises (unless it rains, but then who wants to get up on a rainy day).
Photocells are basically light dependent resistors, which means it is a resistance that changes with the amount of light it receives. Usually, the resistance is very high (100kOhm or more) when there is no light, and falls under 1kOhm in bright light.
Next in line is the IR range sensor. Put 5V through and look at the output to know how far is the closest obstacle ahead. I use a 10-80cm sensor, which means you can detect something up to 80 cm away, it is very accurate between 10 and 30cm, but doesn't give anything good under 10 cm. For a small robot, you can easily plan a turn when an obstacle is 10-15 cm, thus avoiding contact. Very convenient and easy to use.
A prototype of wall-avoiding robot using IR distance sensor
Finally, other great robotic sensors are the accelerometers. An accelerometer calculates the acceleration of your robot, of course, but mostly it gives the global force applied on your robot (remember what Newton said about that?), and that is powerful. That's the key to making a humanoid robot stand and walk, as well as building a Segway-like balancing robot. Accelerometers are also an essential tool in controlling 3D robots, by that I mean aerial or submarine robots.
While you may not need it yourself, force sensors are often used in humanoid robots to get the force under the feet. It helps them staying more stable. They can also be used in the hands or on robotic arms to make sure you don't brake anything.
Even more advanced than force sensors are torque sensors: they measure how much "twisting action" is applied to an articulation. This is used to create compliant robots that can naturally adapt to their movements being blocked. If you block the arm of a normal robot trying to throw a ball, it could try so hard that it brakes its arms following the initial trajectory. But with torque sensors, the robot could detect that something is blocking and stop its movement.
Now, robotic sensors that you can use at home, are microphones and cameras. Actually, you can buy some kits that allow speech recognition on embedded platform, or some embedded cameras with processing unit. But it's really a lot more fun when you have a PC or embedded computer to control it all.
With a microphone, you can detect sound, even where they come from, you can have some basic voice control, and music detection for autonoumous dancing.
With cameras, you can detect objects, if you have 2 you can calculate distances. You can also detect and recognize faces very easily, which means your robot can recognize the members of your family. Now, you can even use Kinect to control your robot with gestures, and have a very useful 3D images. Kinect makes it easy to create 3D maps of your environment, avoid obstacles and recognize objects from shape AND color.
I hope this list of robotic sensors have sparkled some project ideas and helps you to start building your robot. Actually, each sensor is usually very easy to use. You really just need to know that they actually exist and start finding some use for them.
Other links that may interest you:
Learn the basics of robot programming. What's the difference between a PC and microcontroller? How can you structure the AI of your robot like a pro? The answer is here.
How to choose your first robot?. Need help sorting the options for your first robot? This is the best guide to choose it.