Emitter Resistance in BJT Amplifier
This is the basic circuit you seem to be asking about. It's known as a common-emitter amplifier with emitter degeneration:



The two resistors have totally different functions.

RC is totally fundamental to the operation of a common-emitter amplifier. It determines the voltage gain of the amplifier. 

If you removed it, you simply wouldn't have a working circuit, because there'd be no path for current to flow through the collector of the BJT. If you removed it and replaced it with a short circuit, the BJT would still produce current gain, but the output voltage would always be exactly equal to the V+ voltage, and the circuit just wouldn't be very useful.

Note: In some cases, RC is not present, but the load is connected from the collector to the positive supply, so that the load itself fills the role of RC.

RE, on the other hand, is a little more complicated. This resistor is why we call the circuit "emitter degenerate". Having RE means that an increase in collector current tends to reduce Vbe, which reduces the portion of the input voltage that contributes to gain. This is a form of negative feedback. The main benefits of this is that it increases the range of input bias where the circuit operates linearly, makes the circuit gain more stable if the BJT properties vary, and it increases the input resistance of the circuit.

If you removed RE and replaced it with a wire you'd just have a standard common-emitter amplifier.