If an object is moving with no force acting upon it, then it will move with constant velocity. Note that velocity is a vector, so this statement implies that the object will keep the same speed and the same direction of motion. This directly contradicts the animistic view of motion in which the natural condition of a body is at rest with respect to its surroundings - the First Law says the natural state of a body is moving with zero acceleration, not zero velocity.
Motivation for Concept
On the earth it is hard to remove forces. Objects are always contacting other objects (air molecules if nothing else). Even if object-object interactions are minimized, there is the everpresent pull of gravity. Thus, we are used to changing velocities. On earth, Newton's 1st Law can only be understood as a limiting concept. Imagine a hockey puck kicked on carpet, where friction is significant. It will go straight, but not very far. Now, imagine a hockey puck kicked on a hardwood floor. It will go farther. Finally, kick the puck on smooth, wet ice. The puck will go a very long way. You can imagine that if the perfect frictionless surface could be designed, the puck would move forever.
Statement of the Law
Newton's Statement
"Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed." (The Principia by I. Newton, translated by I.B. Cohen and A. Whitman.)
Modern Statement
Modern texts consider the First Law to be a corollary of Newton's Second Law. When no (net) force acts on an object, its momentum is constant.
If the First Law seems unnecessary to you, it is worth remembering that the possibility of the "action at a distance" of gravity (one of the central ideas of Newton's Principia) was by no means a commonly held belief in Newton's day. The First Law might be viewed as a statement that orbits must be explained by a force exerted by the central body, even if no mediator of such a force is observed.