Antennas are used to transmit and receive radio signals. They can range from a simple single wire in the air or a trace on a circuit board to complex multi-element specialized antennas. Each type having their specific purpose. With the intent on transmitting and receiving radio waves the goal is to provide an antenna that does this the best. That usually means designing the antenna for a specific frequency or frequencies, direction of signal, and polarization of signal.
Designing an antenna for a specific frequency is making sure that the antenna is resonant on that frequency. Meaning at that particular frequency the antenna will transmit or receive the power more efficiently than it would at other frequencies. This is where the relationship of frequency and wevelength are crucial. Typically an antenna system will be the most efficient when it is a multiple of the wavelength it needs to transmit or receive. That is it's resonant point. For instance if you want an antenna that recieves on for instance around 150Mhz which is the 2 meter band then you want an antenna that is either full wave, 1/2 wave, 1/4 wave, etc. A full wave antenna would be 2 meters in length. Basically wavelength equals velocity divided by frequency. The velocity meaning that of light in a vacuum being 300Km/s.
λ = v/f
So for this 300 / 150 = 2
It could be a wire 2 meters in length for like a single vertical antenna or it could be split into 2 wires going opposite of each other as in a dipole antenna. To give you an idea of the many antenna types here are just a few to start:
Monopole - is half a dipole. That or just think of it as a single wire sticking up. This is like the stereo antennas on your car.
Dipole - could be simple dipole, folded dipole, short, quarter wave
Yagi - within this can be a simple driven element, a reflector, and a director all the way up to multiple directors up to around 20 element yagis or more.
Reflector - corner and parabolic
Log Periodic - log periodic, bow tie, log periodic dipole
Aperature - Slot, Cavit-Backed Slot, Inverted F, Slotted Waveguide, Horn, Vivaldi, and Telescopes
Antenna types are selected for various reasons. It could be certain types like Horn antennas are mostly used in the Ghz ranges. Others are selected because of they way they are built to transmit or receive only in a particular direction.
The number of types of antennas seems to grow. Here is a really good list of the types of antennas. Each one you can click on and find out more.