Car won't crank, jumps fine

[ab dsm]

Hello!
So I was at work and I went to turn my car on and nothing happened. It was like the battery was totally dead. No lights, no power, nothing, no sound when trying to crank. I got my girlfriend to come jump my car, no problems there. When I got back home, the car wouldn't start again once I has turned it off. I let it sit, and figured I needed a new battery. After letting it sit, I came back and could turn on the lights, and the car made a sound when I opened the door (usual beeping), giving me good signs. I go to crank it and it's like the whole car is shocked or something, after that, no lights, no power... again.
So I go get a new battery anyways. I go to change the battery, and dismount the hooks that keep it held down, and while I start to disconnect one of the terminals, I get some small sparks, so I think well there must be power, so I tried to go crank the car, and bam, no problem, starts up. Let it run a bit, turned it off. The lights still work after turning off. I figured the car was having one of those days, so called it a night.
Go to start it in the morning, turn the lights on and off, no problem. Go to crank: nothing. Now all the lights don't work again.
What is causing this? It jumps no problem repeatedly (had to twice that night) so I know all my connections are still good, it's only when the car goes to use its own power where it seems to have the problem.
Does this sound like a fuse problem? Clutch switch? (Shouldn't be the starter, it jumps perfectly). Alternator is new.
Any help/guidance very much appreciated!

 

[luv2rallye]

The connection of the cable terminals to the battery are dirty/corroded/bad. Remove terminals, sand battery post and where it connects to the terminal with sandpaper, reconnect tight.

 

[broken_eclipse]

Check to make sure the engine is properly grounded. I usually run some 4ga wire from the throttlebody to the negative terminal or chassis to check.

 

[pauleyman]

It jumps no problem repeatedly (had to twice that night) so I know all my connections are still good,

No your connections aren't good. When you jump start it you bypass all that. Likely your connections at the battery are bad. Clean them. The battery itself could also be bad.

 

[1Gina2G]

Everything above AND:

If your battery terminals are looking really bad, or maybe even if they aren't buy some electrical contact cleaner. Will deffenatley clean the conections better then anything.

buy / use a reliable electric meter. test it on other AC/DC sources to confirm the meter works first. Then check your battery between certain periods like before you start, after your just getting home, to get an idea of how / if the battery voltage is fluctuating. You want the battery to be at 12 volts with the car off and 14 while running. Plus or minus 12v is fine but anything under 11volts is too low really.

All electrical connections must be TIGHT. you can't just have wires barely touching / smashed into the terminals. Best thing to do is to invest in some ringed krimp connectors to make removal & reinstallation of the battery wires easier.