Probably the battery or alternator. It could also be the starter malfunctioning but it is easier to check the cheapest and most accessible components first. Bring your battery to an NAPA Autoparts or similar and they will check it for free or even charge it up again. If it keep dying on you, then you have a problem with the electrical system. Next up on the list is the alternator. Take off the belt, unbolt the alternator and bring it to NAPA or similar again and they will test if it is broken or not. You can change it right there and bolt on the new one. Worst case your electrical system is shot and there is a fault somewhere in the system. Takes some time to find out the gremlin in the system, if you have the patience and the tools (electrical tester, light, etc...) then you might want to do it yourself. Otherwise, if you have exhausted all the options above, your only two choices are replace the wiring harness, (about 70-200 usd depending on car and maker) or bring it to a shop and ask them to figure it out, but make sure you tell them what you did and checked for so that they wont check over what you already replaced/fixed and waste time and money. Good luck.
You lost the 12 volts feeding those lights and maybe the 12v to other ciruits including the engine control computor. Start looking for bad fuses and bad connections.
It could be the starter, keyswitch, neutral lockout switch, wiring connector or any of a number of other problems. Unfortunately you have to just track it down.
The starter relay is integrated into the starter. Have you checked the neutral lockout switch? If the neutral lockout switch is faulty or adjusted wrong the starter will never engage. It could also be that the starter wires are cross-connected. Using a volt meter, make sure you are getting voltage to the starter bendix when you attempt to start the engine. If not, start backtracking to the keyswitch to see where you DO have voltage.
Sounds like starter solenoid needs to be replaced
Dead battery? Loose or dirty battery cables? Bad starter solenoid? Bad starter? Neutral switch?
The starter solenoid, or ignition switch is bad, and either one needs to be replaced.
Check your starter, and all power connections.
check ecm fuse
Battery cable connections may not be tight. The starter could be bad. The starter solenoid could be bad. Check to see if the lights work. If they do, leave them on and try to start the vehicle. IF the lights dim, you have a bad battery connection somewhere or a bad starter.
The Starter, Is Bad.
Did you follow the starter alignment instructions when you replaced the starter? Are you getting oil on the Bendix? That could cause the starter gear to engage slowly. It is possible that the replacment starter has a problem. Also, if the previous starter had the same problem it could have damaged the gear on the flex plate.
the starter isn't getting power or the battery is dead. do some tracing of the power and see where it isn't getting juice
replace the starter
Assuming it started before the starter was replaced, the starter and/or starter wiring are not installed correctly.
Simple, buy a new starter.
Assuming you have checked that the battery cables are clean and tight on both ends and you do hear a click coming from the starter area, I'm going to say the starter/starter solenoid is bad. If you don't even get a click it may be the neutral safety switch.
The starter motor requires a great deal more of electrical current than do ordinary light bulbs. The lights working is not always an accurate measure of the battery condition.
Could be battery, battery connections, starter, starter connections, neutral safety switch. I would check those in that order.