Velocity
Velocity is speed in a specific direction, a vector equal to displacement divided by the time interval.
Formula
\vec{v} = \frac{\Delta \vec{x}}{\Delta t}
Definition
Velocity is speed with a direction attached: "10 meters per second east" is a velocity, while "10 meters per second" alone is a speed. It is the rate of change of position, displacement divided by the time interval, $\vec{v} = \Delta\vec{x}/\Delta t$, a vector whose magnitude is the speed and which can be positive or negative on a number line depending on which way the object moves. Instantaneous velocity is the time derivative of the position vector, $\vec{v} = d\vec{r}/dt$, tangent to the trajectory at every point. In two dimensions, $\vec{v} = \langle v_x, v_y \rangle$ with $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$. Velocity is frame-dependent and transforms between inertial frames by vector addition, which is Galilean relativity.
Example
Two trains each travel at $25\ \text{m/s}$, one heading north and the other south: same speed, different velocities. A swimmer who goes $50\ \text{m}$ down the pool in $25\ \text{s}$ and $50\ \text{m}$ back in $30\ \text{s}$ has an average speed of $100 \div 55 \approx 1.8\ \text{m/s}$, but her average velocity is $0$ because her displacement is zero. In two dimensions, a boat that points north at $4\ \text{m/s}$ relative to the water while a river flows east at $3\ \text{m/s}$ has a velocity relative to the shore of $\langle 3, 4 \rangle\ \text{m/s}$, magnitude $5\ \text{m/s}$, directed $53^\circ$ east of north.
Key Insight
If velocity changes, either the speed changed or the direction changed, so a car rounding a curve at a steady $40\ \text{km/h}$ still has a changing velocity. On a position-versus-time graph, velocity is the slope, and a negative slope means motion in the negative direction; speed can never be negative, but velocity can. Constant speed with changing direction is still acceleration, which is why an orbiting satellite is accelerating toward Earth the whole time. That centripetal acceleration is exactly what gravity supplies.