Normally you want a half wave which is known as a "dipole". A quarter wave is usually used in vertical applications because the ground plane or earth ground if used will make up the other quarter to give you an effective dipole. So why a dipole? Because if you follow the voltage to current phase relationship along the antenna, at it's resonant frequency we find the voltage will be at it's lowest impedance at the center. By feeding the center we have fairly good balance. If the impedance is low than it benefits both transmitter and receiver. In the case of a receiver, the low impedance keeps the rf front end from picking up allot of junk, noise, when we match to the antenna impedance. From the transmitter standpoint the low 50 ohms impedance helps to prevent possible feedback from getting back into rf power amplifier stages.
We DONT. A resonant antenna should be 1/2λ! A 1/2λ antenna is called a dipole. It can be end-fed, or center-fed.
A 1/4λ antenna will work only if it is mounted orthogonally to a ground-plane which is ~1/2λ by 1/2λ in area mo. Think about a 1/4λ whip antenna mounted on the roof of a car, for example. The metal ground plane (car roof) must be at least 1/4λ in radius from where the whip is mounted. Another example is here.
An end fed 1/4λ without a ground plane works about as well as a wet noodle.