Primary ignition circuit not hooked up, coil hot in start mode but not in run mode.
General karthief is right, you have some missing wiring...
Engine fires briefly, but dies immediately when the key is released to the Run position. Crank the engine & when it fires off, pull the small push on connector (red/blue wire) off the starter relay (Looks like it is stuck on a screw). Hold the switch in the crank position: if it continues to run there is a problem with either the ignition switch or TFI module. Check for 12 volts at the red/green wire on the coil with the switch in the Run position. Good 12 volts, then replace the TFI.
See the Ignition switch wiring diagram for more information on the ignition wiring fuse link because it is the next thing to be tested. You will need a Multimeter or DVM and know how to use the Ohms function to check continuity between the red/green wire on the ignition coil and the red/green wire on the ignition switch. Make sure that the ignition switch is in the off position when you do the check. You should see less than 1 Ω (Ohm) between the red/green wire on the coil and the red/green wire on the ignition switch. More than 1 Ω means that the fuse link may have blown open and needs to be replaced. If you get 1 Ω or less means the fuse link is OK and the ignition switch is bad.
The following are diagrams courtesy of Tmoss & Stang&2birds
Complete computer, actuator & sensor wiring diagram for 88-90 Mass Air Mustangs
Complete computer, actuator & sensor wiring diagram for 91-93 Mass Air Mustangs
See the following website for some help from Tmoss (diagram designer) & Stang&2Birds (website host) for help on 88-95 wiring; http://www.veryuseful.com/mustang/tech/engine/ Everyone should bookmark this site.
TFI module wiring for 94-95 Mustang GT
http://www.veryuseful.com/mustang/tech/engine/images/Mustang-94-95-IgnitionControlModule.gif
Complete computer, actuator & sensor wiring diagram for 94-95 Mustangs
Complete computer, actuator & sensor wiring diagram for 91-93 Mass Air Mustangs
Complete computer, actuator & sensor wiring diagram for 88-90 Mass Air Mustangs
Ignition switch wiring
Fuel, alternator, A/C and ignition wiring
O2 sensor wiring harness
Vacuum diagram 89-93 Mustangs
HVAC vacuum diagram
TFI module differences & pin out
Fuse box layout
Mustang 5.0 Lights and Radio schematic, by TMoss:
http://www.veryuseful.com/mustang/tech/engine/images/mustangFoxLights-Radio_diag.gif
87-92 power window wiring
93 power window wiring
T5 Cutaway showing T5 internal parts
Visual comparison of the Ford Fuel Injectors, picture by TMoss:
Convertible top motor wiring
http://www.veryuseful.com/mustang/tech/engine/images/mustang88VertTopMotorCkt.gif
Engine mounted fuel injector harness
Location of the TPS, IAB, and the 10-pin connectors on a 5.0, picture by TMoss:
Starter circuit
Alternator diagram for 94-95 Mustangs.