What is Delta-V?

Delta-V: In spacecraft flight dynamics, Delta-V (generally written as Δv) is shorthand notation for the change in velocity required to execute an orbital maneuver.

Hohmann Transfers are a good examples of how Delta-V is used in orbital mechanics.  A Hohmann Transfer is an elliptical orbit used to transfer between two coplanar circular orbits.  They are one of the lowest propellant-intensive methods for transferring between said orbits.  For example, a spacecraft in low Earth orbit (LEO) can use a Hohmann transfer to get to the moon's orbit (let's just say it's delivering coffee).  The LEO and the moon's orbit would be the two aforementioned circular orbits.  This is illustrated below:

Where:

r1 is the radius of the LEO
r2 is the radius of the moon's orbit
Δv1 is the change in velocity required to put the spacecraft on the Hohmann transfer orbit
Δv2 is the change in velocity required at the moon to insert the spacecraft into the moon's orbit.
 
The spacecraft ignites its engines at Δv1 in LEO to increase its velocity.  This increase in velocity causes the apogee of the spacecraft's orbit to increase.  Eventually, the apogee of the orbit reaches the radius of the moon's orbit.  That velocity change is Δv1.  The spacecraft shuts down its engines after the apogee reaches the desired radius, it is now in the Hohmann transfer orbit.  After coasting in the transfer orbit, the spacecraft reaches the moon's orbit.  It again ignites its engines (at Δv2 this time) to increase its velocity.  This increase in velocity causes the perigee of the spacecraft's orbit to increase.  When the required Δv is achieved (Δv2), the spacecraft shuts down its engines.  There is now a spacecraft loaded with coffee coasting in the same orbit as the moon!