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Light
Visible light is also electromagnetic radiation. When a rainbow splits visible light into different colors, it's really separating out the different wavelengths. Blue light has a shorter wavelength than red light, while yellow and green are between them. In the same way, infrared light is like red light with an even longer wavelength, and radio waves are far longer still. So you can think of infrared radiation as a "color" with a wavelength about twice as long as visible light, and radio waves as a "color" with a wavelength about two million times as long as visible light. If you could see the waves as they were travelling you'd see they look exactly the same except for their wavelength.
Waves and Energy
Many years of experiment and calculation have gone into examining the nature of electromagnetic radiation. As far as your remote control goes, there are a few important conclusions. First, radio and infrared waves both move at the speed of light. Second, because radio waves are longer it takes longer for the wave to move from peak to valley and back to peak. Third, if you could take the tiniest little packet of radio waves and compare it to the tiniest little packet of infrared radiation, you'd find the radio packet carried only one-millionth of the energy of the infrared packet. That's important because a packet of electromagnetic energy will only interact with objects that are about at that same level of energy.
Electromagnetic Interactions
When an electromagnetic wave reaches an object, there are only three things that can happen: it can reflect off, it can get absorbed or it can pass right through. The energy of the electromagnetic energy packet is what determines which of those three things will happen. Because radio waves and infrared radiation are so different in terms of energy, they reflect, get absorbed or pass right through completely different types of objects. Radio waves, for example, will pass right through glass, wood, carpet and you. Some infrared waves might pass through some glass, but they won't make it through wood, carpet or you.
Remote Control Differences
Infrared and radio remote controls act completely differently because of their different wavelengths, which cause them to interact with materials differently. A handheld radio remote control can control a model airplane hundreds of yards away. An infrared remote control signal gets absorbed a little bit every time it goes past molecules in the air, so a typical infrared remote won't work much further away than the length of a room.
Infrared remotes send their signal pretty much in a straight line, or a slight cone, so you need to point your remote pretty close to the object you're controlling. Infrared will also reflect off many types of surfaces, so sometimes you can aim your infrared remote so the reflection heads towards the object. Remote control radio antennas are so short compared to their wavelength that the signal spreads out in all directions, so you don't really need to point a radio remote in any specific direction.
Finally, you can use a radio remote control to send commands to the other side of a wall -- as long as it's not made of metal -- while your infrared remote will work only if you can see the object you're controlling.
Control Signals
Radio control and infrared control both work by the same principles. The electromagnetic waves are modified -- the technical term is "modulated" -- to carry information. Although there are some pretty fancy modulation schemes out there, you can think of modulation as something like a bright flash counting as a "one" and a dim flash counting as a "zero." Your computer uses strings of zeroes and ones to carry information; your remote control does the same thing. Because infrared radiation has a shorter wavelength it moves from maximum to minimum a million times faster than radio waves -- which means it can carry information about a million times faster. That's usually not important with a remote control, because the signals are very short.