MOSFET THEORY:
So lets go with the N-channel MOSFET (NMOS) since they are cheaper, more efficient, switch faster, and easier to find than the PMOS.
Your voltage 12V, so you need one with a breakdown source-drain voltage of >12V (try not to cut it too close). Your peltier is rated for 110W, so at 12V, so the max current expected is 110W/12V = 9.2A. So it needs to be able to handle at least that much current.
Since the 555 timer can takea maximum of 15-18V, you should be able to run it directly off your 12V PSU. THis also means that the 555 timer output will be 12V which means you can use MOSFETs that do not have a logic-level gate threshold switching voltage (which are 2.5-3.3-5V or so, vs the typical 10V). It should be pretty easy and cheap to find such a MOSFET.
In an NMOS, the gate pin goes to the output on the 555 timer (duh). The NMOS can only block conventional current trying to flow from drain to source. This means that the source must be more negative than the drain (current flowing the other way will see a diode and will flow right past).
Since the triggering voltage on the MOSFET gate is referenced to the source and the 555 timer output voltage is refenced to ground, this means you shoudl connect the source to ground so that the NMOS can easily surpass the gate threshold voltage (so the driving signal has the same reference as the threshold voltage). ALl this means is that it's easiest to switch an NMOS on and off it is on the low-side than if it is on the high-side (you know what those terms means right?)
ie. if the source voltage is 7V higher than the ground because you connected the transistor closer to +12V than GND (high-side), then the 555 timer will need to output an extra 7V to turn the transistor on which it cannot do).
So connect the source-drain of the NMOS on low-side (closer to ground) with the Souce-drain polarity in the same way. The peltier goes on the high-side (closer to +12V).
Then you just set up the 555 timer to output PWM pulses using the astable circuit with variable resistor (manual or temperature) to control the PWM duty cycle which will switch the MOSFET on and off int he same way.
BTW, in case you haven't already figure it out, a MOSFET blocks current (going from drain-to-source only, the other directions just looks like a diode) when the voltage from gate-to-source is less than Vthreshold, and turns allows current to pass when the voltage is larger so it acts like a one-way switch.
OR BLINDLY DO THIS:
The connection between 555 timer and MOSFET looks like this:
**broken link removed**
like this except that your load is not a motor, but a Peltier device. The circuit I had in mind used output to drive the MOSFET. In this one they use the discharge pin and a pull-up resistor. There are a few ways to do it. Shouldn't be a problem for you to figure out the best one since you have used 555 timers before.
EDIT: The one I had in mind, I was not sure if it kept a constant pulsewidth and pulse frequency as the duty cycle changes (I'm pretty sure it didnt.) The provided circuit does. Keep the frequency in the range of the 555 timer and MOSFET. Maybe keep it low since it's just a peltier and doesn't really matter. I'd choose 1kHz.
IMPROVEMENTS:
Also, remember how I said thermistors are non-linear? I never thought about using them as a voltage divider for both charging and discharging like the circuit provided (which also keeps the pulse-width constant with changing duty cycles). Just a quick thought which is unsubstantiated but you *might* be able to make it linear if you use two thermistors as a voltage divider (or something similar) so that their non-linearities cancel each other out. you'd have to look into it more.
You might also want to put a fixed resistor somewhere (one at the end of the potentiometer will produce a minimum or maximum achievable PWM duty cycle depending on which side it is on. One at the center-tap of the variable resistor will produce equal max/minimum limits on the duty cycle. The upper limit is because if you realize that your PSU cannot handle the full Peltier current, then you need to physically limit it somehow so you (or the thermistors) don't acidentally turn it up too high. There might also be a minimum duty cycle required which would cause your CPU to overheat if you went below it.
For the record, I think you should a fan rather than the Peltier and if you really needed below ambient cooling, then I think you should use water cooling or a vapour-phase system. I just like the 555-timer PWM circuit.