Speed is an attribute which every unit has. It measures how fast a unit can move and it's represented as a shoe with wings on it. The advantages of speed varies with where it's used.
Ground units in ages before gunfire could dodge arrows flying at them by moving out of the way when launched, this is easier with mounted units, as they typically start with 16 units of speed. The hit-and-run tactic involves mounted archers constantly moving to and from the enemy in hopes of evading as much damage as possible and causing the most damage possible. When guns are introduced, they never miss, and hit their target instantly, which renders dodging useless.
Aircraft benefit from speed as the faster they go, the less flight time (fuel) is needed to go to their destination. Therefore, increasing the "range" of your air units. Aircraft can also evade some damage from anti-air (AA) because the faster the plane is, the faster it can fly past (or retreat from) the AA. Slower aircraft will take more gunfire (or missle), damaging the already fragile airplane.
Tip: this isn't an excuse to not seek out and destroy as much AA as possible before sending your air in.
All units benefit from increased speed in that they reach their destination sooner.
Group speed (or "the speed glitch") is one of the few legal glitches that can be performed in the game. The glitch involves using your army (of slow units) with faster units, such as mounted units. Naval units and aircraft do not have group speed, and do not effect the speed of other units. The speed of all units in a group is averaged to form a new group speed. So for example, a group of 10 Ballistas and one Cavalry Archer will just slow your one CA down a lot. It's best to have a group with a majority of fast units and only a few slow units (i.e. 30 Dragoons with 16+ speed, and 10 Bombards). This is one of the most commonly used glitches, and new players may not even be aware that they're using the glitch themselves unintentionally.
To perform, select all of your army you wish to send together, and move your army to a destination by HOLDING down the right mouse button and releasing it (a yellow arrow should appear). This is clear when you see a bunch of siege moving at the same speed as your cavalry.