There are many ways how you could possibly define different types of robots. As I have seen the possible divisions vary widely. The main reason of these differences is that different tutors often tend to have different views on what should be taught under "robotics".
For example - some tutors that teach robotics usually focus mainly on industrial robotics, neglecting service robots completely. Therefore when talking about types of robots they usually talk about types of industrial robots. There is a strong reason for this though - the vast majority of robotics engineers will have to deal mostly with industrial robots in their careers.
Nevertheless, industrial robots are not the only ones. Therefore, as I see it when dividing robots into types this division should be broad enough to include everything that can be understood as a robot.
There are two possible ways how this could be done. First, you could divide robots into types by their application and second - by the way they move (or don't). I acknowledge that there are other possible ways how to divide robots into types but in my opinion these two are the best. Also, I prefer to use both these classifications together. This way two questions about a robot would already be answered - "What it does?" and "How it does it?"
*Industrial robots - Industrial robots are robots used in an industrial manufacturing environment. Usually these are articulated arms specifically developed for such applications as welding, material handling, painting and others. If we judge purely by application this type could also include some automated guided vehicles and other robots.
*Domestic or household robots - Robots used at home. This type of robots includes many quite different devices such as robotic vacuum cleaners, robotic pool cleaners, sweepers, gutter cleaners and other robots that can do different chores. Also, some surveillance and telepresence robots could be regarded as household robots if used in that environment.
*Medical robots - Robots used in medicine and medical institutions. First and foremost - surgery robots. Also, some automated guided vehicles and maybe lifting aides.
*Service robots - Robots that dont fall into other types by usage. These could be different data gathering robots, robots made to show off technologies, robots used for research, etc.
*Military robots - Robots used in military. This type of robots includes bomb disposal robots, different transportation robots, reconnaissance drones. Often robots initially created for military purposes can be used in law enforcement, search and rescue and other related fields.
*Entertainment robots - These are robots used for entertainment. This is a very broad category. It starts with toy robots such as robosapien or the running alarm clock and ends with real heavyweights such as articulated robot arms used as motion simulators.
*Space robots - Id like to single out robots used in space as a separate type. This type would include robots used on the International Space Station, Canadarm that was used in Shuttles, as well as Mars rovers and other robots used in space.
*Hobby and competition robots - Robots that you create. Line followers, sumo-bots, robots made just for fun and robots made for competition.
Now, as you can see there are examples that fit into more than one of these types. For example, there can be a deep sea exploration robot that can gather some valuable information that can be used for military purposes.
Also, I have seen that a division into two types is used, accordingly - industrial and service robots. However, I can not see how a Mars exploration rover fits into one of these general types. Therefore I have used "service robots" in a narrower manner. In my version a term "service robots" serves as "others". This is basically a type where robots that don't fit into other types should fall in.
As you can understand, robot's application alone does not provide enough information when talking about a specific robot. For example an industrial robot - usually, when talking about industrial robots we think of stationary robots in a work cell that do a specific task. That's alright, but if there is an AGV (Automated Guided Vehicle) in a factory? Its also a robotic device working in an industrial environment. So, I propose to use both these classifications together.
So there are:
1.4 SCARA robots
1.5 Articulated robots (robotic arms)
1.6 Parallel robots
2. Wheeled robots
2.2 Two-wheel robots
2.3 Three and more wheel robots
3. Legged robots
3.1 Bipedal robots (humanoid robots)
3.2 Tripedal robots
3.3 quadrupedal robots
3.4 hexapod robots
3.5 other number of legs
4. Swimming robots
5. Flying robots
7. Swarm robots
Wondering about others? Yes, there are others. For example snake-like robots. There are many fields of research that deal with different innovative types of robots. Someday they will be very useful. However, by now I'll put them under the "others" type.
Of course, nothing of this is carved in stone, especially in robotics where everything changes almost monthly nowadays. Still, in my opinion these types of classification do their job well enough.
As you may have noticed, I use the classification by application as the main outline of this site. So if a certain type of robots interests you its possible to find more info about it on the left hand navigation menu if its there.