By soft reset, I mean reboot the game directly, without returning to the BIOS screen. (similar to how on a Wii, when you push Reset while running a GameCube game, it resets the GC game but doesn't drop you back at the system menu)
A "vector" is a 2-byte pointer to each of the three functions in the CPU's address space.
NMI is "non-maskable interrupt". An "interrupt" is a piece of code that is named because it makes the CPU interrupt whatever routine it was working on (needing a special instruction, RTI, to tell it to resume what it was doing previously to the interruption). "non-maskable" means there isn't a CPU instruction that can explicitly disable it (not affected by CLI, which I think is the relevant instruction). In this case of the NES the NMI is initiated by the PPU. (the CPU can write a command to tell the PPU to disable the NMI. However, it is NMI since the CPU cannot itself disable it.)
The PPU forces that routine to run once each frame, which is often how timing is done on the NES. Typically NMI will also run code to check for pending VRAM/palette write requests (ideally first since it's the most time-critical) as well as sound driver and controller input checks.
Reset is, as may have been implied, the start of the game code. Pushing Power or Reset sets the CPU to that instruction. Some Konami FDS games will call the Reset function of the FDS BIOS directly, as an antipiracy measure to effectively endlessly reboot the console if it detects the game has been hacked.
IRQ is a standard interrupt. I think it can be initiated by the BRK instruction but I've never seen it done intentionally (I've only seen it before because I recall it is opcode 00, making it a common thing to happen when debugging bad code.) The stock NES hardware doesn't use it but some of the more advanced mappers (such as MMC3) use it for some of the more advanced functions (like split-screen scrolling).
Oddly, the Master System uses NMI for the Pause button and makes Reset function like a gameplay button the game can ignore if it feels like it. Sounds like they should've been handled the opposite.