In general, a counter is a device which stores (and sometimes displays) the number of times a particular event or process has occurred, often in relationship to a clock signal. In practice, there are two types of counters:

up counters which increase (increment) in value
down counters which decrease (decrement) in value

In electronics, counters can be implemented quite easily using register-type circuits such as the flip-flop, and a wide variety of designs exist, e.g:

Asynchronous (ripple) counters
Synchronous counters
Johnson counters
Decade counters
Up-Down counters
Ring counters
Each is useful for different applications. Usually, counter circuits are digital in nature, and count in binary, or sometimes binary coded decimal. Many types of counter circuit are available as digital building blocks, for example a number of chips in the 4000 series implement different counters.

Asynchronous (ripple) counters

Asynchronous Counter created from JK flip-flops.The simplest counter circuit is a single D-type flip flop, with its D (data) input fed from its own inverted output. This circuit can store one bit, and hence can count from zero to one before it overflows (starts over from 0). This counter will increment once for every clock cycle and takes two clock cycles to overflow, so every cycle it will alternate between a transition from 0 to 1 and a transition from 1 to 0. Notice that this creates a new clock with a 50% duty cycle at exactly half the frequency of the input clock. If this output is then used as the clock signal for a similarly arranged D flip flop (remembering to invert the output to the input), you will get another 1 bit counter that counts half as fast. Putting them together yields a two bit counter:

cycle Q1 Q0 (Q1:Q0)dec
0 0 0 0
1 0 1 1
2 1 0 2
3 1 1 3
4 0 0 0