Just think of a vector in a micro as a "pointer" to a fixed address. That is, an interrupt will always trigger to a fixed address to start the interrupt code. These are usually hardwired in the micro.
For example, look at the memory map of the micro. In an example, the 16F887 datasheet, Section 2, "Memory Organization", it shows the "Interrupt Vector" to be at address 0004H. Whenever an interrupt is triggered, the PIC will jump to that address, which usually contains a "jump" instruction to your written interrupt code.
The "Reset Vector" is at 0000H, which is the power on/reset starting address, which also (usually) contains a Jump instruction to the beginning of the code.
Other micros may have several "vectors" depending on functions or other hardware features