When referring to transistors what is meant when the datasheet says:
Collector Current—Continuous: 600 mAdc
Or
Collector Current: 15 Adc
(2 different models)

Is that (milli) Amps Direct Current?

Recently started an electronics class and was reading ahead, now its bugging me. That's what I get for studying, :) Thanks!

Yes.
The transistors collector requires a constant 600 MilliAmps of DC current, or 15 Amps of DC current.

Computers, and transistors, all use Direct Current.

The Power Supply (SMPS for today's computers), converts AC electricity into low DC voltages for the computer.

The transistor example that uses 600 MilliAmps is a small one.
The one using 15 Amps, (Amperes) sounds like a MOSFET.
More specifically a Power MOSFET,

Power MOSFET - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Power_MOSFET)

You may find these two websites will help increase your knowledge.
One based on simple info, the other more to what you are studying now,

1) HowStuffWorks "Computer" (http://computer.howstuffworks.com/)

2) Uncomplicating the Complicated | Hardware Secrets (http://www.hardwaresecrets.com/)

More specifically for Number 2 above,

CPU | Hardware Secrets (http://www.hardwaresecrets.com/page/cpu)

How a CPU Works | Hardware Secrets (http://www.hardwaresecrets.com/article/How-a-CPU-Works/209)

In my opinion, the thinking is to compare how a Central Processing Unit works, and use your new found transistor info to compare, and apply.

Regards,
joecoolvette