Waves are everywhere! In the air, in water, even in solid objects, visible and invisible, waves carry information - sometimes useful (like speech or TV data) and sometimes purely incidental (waves in the sea, or noise in the street).
Visible waves are probably the easiest to think about, and many types of mechanical oscillation generate visible waves. Have a look at the example below: a group of people jump up and sit back down, in sequence...
...note that something (we could call it information) travels from left to right, but the people themselves only stand up and sit down - no-one swaps seats. Here the medium in which the wave propagates is 'people'...no people=no waves. The wave is mechanical - no electrical or acoustic stuff is happening, just a physical movement. In general, mechanical waves are transmitted by vibrating particles (in this case the particles are the people), but although they oscillate (stand up / sit down) the particles do not move from their normal position.
In the mexican wave example shown above, the movement of the people is vertical (up and down) whereas the wave travels horizontally (from left to right). This is a transverse wave - the vibration direction is perpendicular (at right angles) to the direction in which the wave travels or propagates. Transverse waves are very common - waves in water and on strings are two good examples..
A transverse wave can also be set up on a slinky spring - have a look at this animation: