Car starts in neutral, and shifts into gears when turned off but wont engage into gears outside of neutral when car is running?

Start with checking the fluid level. Make sure to follow the owner's manual instructions. Some cars require the engine to be running and in a neutral. If the level is correct see if it will go into another gear. Check the linkage/ cable to make sure it is actually functioning and shifting the transmission.

Unfortunately with automatic transmission your options are limited.

hangon. your leaving something out. myguess is you had the ebrake pulled so the car didn't roll when it was in gear. and now the car wont move at all or what? you left alot of important info out. if it is running and wont move at any point you broke something in the trany or a driveshaft. you should not be able to shift the trany into any gear if the engine is running. the clutch is turning and unless the clutch is disengaged you wont be able to shift. knowing that i am guessing you broke it good and need a new trany

Brake ON and shift gear lever to N - neutral and try to start.

It can be a bad neutral safety switch.

sounds like a broken shift cable. (the connection between the shifter and the gearbox) common symptoms are the shifter move easier than normal.and does not move to any gears (not even park or neutral).

Do you feel any resistance in the gear selector when shifted through gears?

Sounds like a problem with the brake light switch. Do the brake lights work? It is a problem with the shift interlock system. It is controlled by the brake light switch on the brake pedal bracket.

CHECK BRAKE LIGHT SWITCH FUSE AND THE BRAKE TRANSMISSION SHIFT INTERLOCK SOLENOID FUSE.IF THEY BOTH CHECK OUT OKAY.YOU HAVE A FAULTY BRAKE LIGHT SWITCH OR FAULTY BRAKE TRANSMISSION SHIFT INTERLOCK SOLENOID.

if you can move it into gear when its not running, id say clutch,
but if it wont go in at all, then Im thinking shift forks or linkage

follow instructions in manual for SHIFTLOCK RELEASE

Turn off car.
Engage Parking Break
Take ignition key and put into shift lock release slot (near gear shift)
Push down on key and at same time MOVE gear shift to neutral.
Remove key, start car as normal and disengage parking break...

good to go.... mechanic can replace/repair if this is an ongoing issue: $85 I read somewhere...but heard it's not hard to do yourself for $35 someone else wrote.

Goodluck