First off: Missiles typically do not "chase down" fighter jets as misrepresented in movies. An effective missile shot will have the missile reaching the aircraft with a much greater speed (like twice as fast). Aircraft perform defensive maneuvers against missiles, but it is not similar to dogfighting, as you might imagine. They are typically fast, high-G slices (descending turns) that use the missile's high speed against it. (A higher speed means a greater turning radius, which means the missile may not be able to turn into the fighter, but will instead overshoot.)
A missile, when launched at other than point-blank range, will immediately fire its rocket motor and gain as much altitude as possible. Once the motor burns out, the missile then glides to its target. The potential energy it stores in the form of its higher altitude can be spent on either gliding a greater distance to a target, or maneuvering to track a maneuvering target. It has a fixed amount of energy to spend on either of these, both of which decrease the missile's altitude and speed. In order to be lethal, the missile must have enough energy to arrive at the target's location with sufficient altitude and airspeed to maneuver in the terminal phase and explode within its lethal radius.
Fighter pilots continually evaluate a missile's Pk (probability of kill) against what actions the enemy is likely to take against the missile. To help them judge whether a shot is good, the aircraft displays symbology indicating the missile's effectiveness against different "types" of targets (maneuvering, non-maneuvering, etc.). This symbology appears as a DLZ (dynamic launch zone), a vertical range scale that looks like this:
On the left is a caret that indicates where along the range scale the target currently is. On the right are the lethal ranges for various types of targets. Let's see what they each mean:
Raero (aerodynamic range) is the maximum range the missile can kinetically fly. The missile would burn to its maximum altitude, and then glide its maximum distance with no maneuvering. A missile launched at this range would only be effective against a target that flew in a straight line towards the fighter.
(Note that these ranges represent the range between the fighter and the target at the moment in time when the missile impacts [so-called F-pole range], not when the missile is launched. So, Raero applies to targets that head directly towards the fighter, not targets that remain stationary somehow.)
Ropt (optimal range) is similar to Raero but leaves the missile with enough energy to maneuver in the terminal phase against a defending target (so-called high termination criteria). So in other words, a shot at this range would be effective against a target that heads directly towards the fighter, then performs defensive maneuvers at the terminal phase.
Rpi (probability of intercept) is the same as Ropt but does not require the missile to loft (burn to a higher altitude). A target at this range allows the missile to burn straight ahead towards it.
Rtr (turn-and-run) is the maximum range a target can immediately turn and fly the opposite direction and still be reachable by the missile.
Rmin is the minimum range where the missile can be launched, acquire its target, and explode without posing a threat to the launching fighter.
So, as you can see, if the caret is between Rmin and Rtr (the no-escape zone), you can feel comfortable that any missile you launch is guaranteed to reach its target, no matter what the target does. Of course, that's no guarantee it will hit its target: The target could spoof the missile with countermeasures, or outmaneuver it with a well-executed slice, but the missile will at least be capable of reaching the target no matter what.
The caret as it's positioned now indicates that the missile will most likely not be effective if the target reacts in any way other than continuing to fly straight towards the fighter (which is unlikely). Above the DLZ is a figure indicating that the pilot can help improve the Pk by performing a 35-degree pitch up to save the missile from having to do that portion of the loft maneuver itself.
OK, so to actually answer your question now... As you can tell, "it depends."
How far away was the missile fired? What kind of missile was it? There are lots of missiles with differing thrust and power capabilities. What kind of maneuvering did the target do in the initial and terminal phases of the missile's flight? What kind of maneuvering was required by the missile to perform its loft, and correct its heading? How much energy (airspeed and altitude) did the launching fighter have when it launched the missile?
The shape of the DLZ on the HUD changes constantly as the fighter and its target change their headings, airspeeds, and altitudes.
So, an example. The most common medium-range air-to-air missile is the AIM-120 AMRAAM. The current version is the AIM-120C-7, but data on that missile is classified. We know a bit more about the C-4, and can extract some typical numbers about its performance.
For an AIM-120C-4, you can expect Raero to be around 60 miles, and Rtr to be around 30 miles. These are incredibly ballpark numbers and can vary wildly depending on the specifics of the situation. Remember that these are also F-pole ranges, so a value of 60 represents a range r such that, at the two aircrafts' closure rate, their range would be 60 miles at missile impact.