What voltage version of the Pro Mini do you have. The CR2032 will not directly power the 5V version (legally) but will be okay for the 3.3V version assuming your not drawing to much current.
You do not connect to RAW as this is the feed to the on board voltage regulator. You should connect to the VCC pin instead but be aware of the caveat that driving the voltage to high on this pin will damage your Arduino.
Am I correct in saying that I connect the Positive of the battery to the RAW connection on the Pro Mini?
No, you shouldn't directly connect that battery to the Pro Mini (5V version as you didn't specify a 3V3 version). You need a step-up converter. Use one that outputs 5V then don't connect to RAW but to the pin labeled "5V".
Keep in mind that a CR2032 hat a capacity of about 0.6Wh so the Pro Mini won't run very long on it (depends on additional load and sleep strategy).
There is no problem powering a Pro Mini with a CR2032 battery, except that you need to make sure that the clock speed is set to 8 MHz or lower. The processor is not guaranteed to run at 16 MHz using ~3V.
If you have the 5V version, you should do that by changing the fuses to 8 MHz internal RC while powered by 5V. If you have the 3.3V version, it should already have an 8 MHz crystal or resonator. Google for instructions.
To power the Pro Mini using a CR2032, connect to the Vcc (or 5V, not RAW) pins and ground. Don't expect the battery to last more than a couple of hours powering LEDs.
You will find an outstanding tutorial on power saving techniques and sleep modes here. The Pro Mini can run for years on a CR2032 battery, if you do it correctly (no LEDs, remove on board regulator, sleep, etc.).
The project will only have to drive three LED's and a barometric pressure sensor.
So just so I'm clear...
It's positive -> VCC and Negative -> GND?
Not quite. If you are using a power source that does not need voltage regulation (such as a CR2032) then it can be connected directly to the RAW pin. Any power to the VCC pin goes through the regulator, which will drop some voltage even if it is below the output of the regulator (eg, the arduino might only see 2.8V if there is 3.0V at VCC, even though it is a 3.3V regulator.)
Edited after reading below... Sorry for any confusion
(drag mouse from here...
(To me it makes more sense that the RAW pin goes straight to the MCU, as in a RAW direct connection to the MCU, not regulated, and that VCC should be the "system supply voltage", not the board voltage, but this is just semantics... Why couldn't they label it Vin and 3V?, but I digress...)
...to here).
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The arduino should be happy running between 3.2 and 2.5 volts as the battery gradually dies.
As a further suggestion, look into power saving techniques for the Arduino. Otherwise it won't last more than a day on a coin cell. As a side note, I've made an arduino wrist watch that runs on a CR2032 and also logs the temp and pressure over the day. I can get several months of use on a single battery using proper power saving techniques.
If you are using a power source that does not need voltage regulation (such as a CR2032) then it can be connected directly to the RAW pin.
Any power to the VCC pin goes through the regulator.
FIGHT! FIGHT! FIGHT!
Hmmm.....
This woulnt be confusing to the OP .... AT ALL.
........ now would it?
Directly from the STORE PAGES:
Power
The Arduino Pro Mini can be powered with an FTDI cable or breakout board connected to its six pin header, or with a regulated 3.3V or 5V supply (depending on the model) on the Vcc pin. There is a voltage regulator on board so it can accept voltage up to 12VDC. If you're supplying unregulated power to the board, be sure to connect to the "RAW" pin on not VCC.
The power pins are as follows: RAW For supplying a raw voltage to the board. VCC The regulated 3.3 or 5 volt supply. GND Ground pins.
John_S... you might want to update your post.
RAW = GOES THROUGH REGULATOR (ie: voltage is too high needs to be regulated down to 5v)
VCC = NO REGULATOR (ie: voltage is already at correct 5v/3.3v level and needs NO REGULATION)
It is a waste of power to use the RAW input, under ALL circumstances, because it goes through the regulator.
The Pro Mini can and should be powered directly through the 5V (Vcc pin) if using voltages less than 5V. In a permanent battery powered (<5V) installation, remove the regulator entirely with a swipe of a hot solder pencil.