Ok ok I swear after this I'm done asking questions. It's just that I'm trying so hard to understand all of this, but sometimes it just doesn't make sense to me! So here we go:

Question

How many J of energy must be removed when 124.0 g of steam, at a temp. of 167.0 C, is cooled into 124.0 g of ice at 0 degrees C? Take the specific heat of steam to be 2.1 kJ/(kg x K).

Again, I know that the specific heat formula is involved, but I just can't make a connection!

You don't know the final temp, so make a table of heats:
PartA
heat released when steam cooled to 100C
heat released when steam condenses at 100C
Heat released when condensed steam cools from 100C to Tf
Part B
Heat ice absorbs when melted.
Heat melted ice(water) absorbs warming from 0C to Tf

Part A has to equal PartB. Solve for Tf.

Now, recompute part A knowing Tf.

This is done in stages to move T of steam to boiling temp, then condense the steam, then move water from 100 to 0, then freeze the water at zero.

q1 = heat removed from 124.0 g steam to lower the temperature from 167.0 to 100.0 degrees C (but it's still steam)=
mass x specific heat steam x (Tf - Ti) where Tf is final T and Ti is initial T. This works out to be
q1 = 124.0 g x 2.1 J/g*K x (167.0 - 100) = xx J.
Note I changed 2.1 kJ/kg*K to 2.1 J/g*K

q2 = heat removed from steam at 100.0 degrees C to condense the steam to liquid at 100.0 C.
q2 = mass x heat of vap = 124.0 g x 2260 J/g = yy J.

q3 = heat removed to move the temperature of liquid water at 100.0 C to zero C (but it's still liquid water).
mass x specific heat water x (Tf-Ti).
124.0 g x 4.184 J/g*K x (100.0 - 0.0) = zz J.

q4 = heat removed to freeze the water.
mass x heat of fusion = 124.0 x 334 J/g*K = ww J.

Check my thinking. Check the numbers. Make sure I have heat vap and heat fusion correct. Make sure units match.

Note that moving the temperature from one T to another always has the formula
mass x specific heat x (Tf - Ti) while
changing state always has the formula
mass x heat fusion OR heat vap. So you can go up and down the scale but do it in stages and you use only the two formulae above.

It appears to me that the final T is known and it is zero degrees C. From the problem.

Oh wow, I was doing that WAY wrong. But I see now that it is really quite a few more steps then I realized before. It really helps me when you break it all down like that.

I can't even thank you enough for all of your help tonight. You have been so patient, and I appreciate it. :)

Lindsay--
You need to be aware that Bob Pursley and I interpreted the problem differently. His interpretation is that the steam was directed into a bucket of ice and he solved for final T. I interpreted the problem as a simple remove the heat problem from steam to freezing the water into ice. In my interpretation, there was no ice at the beginning. Therefore, you received two answers, both correct, from slightly different perspectives.