Hi Brad and welcome to the electronics point forums
You can do this with a 555-based oscillator driving a transistor or MOSFET.
Have a look at the basic 555 oscillator. It's the first diagram on this page:
http://www.electronics-tutorials.ws/waveforms/555_oscillator.html
You would need to replace the two resistors with potentiometers, to make the high and low periods adjustable. R1 controls the longer period - the ten minute delay. R2 controls the shorter period - the 15 second run time. (Actually, R2 also affects the delay between run periods, though it doesn't contribute much compared to R1 because it's a much lower value.)
Ten minutes is quite a long time for a 555 oscillator. You would need to use a high capacitor value for C1. Assuming you use a 1 megohm potentiometer for R1 (the highest value that's widely available), and you use a 1000 uF (microfarad) capacitor for C1, the maximum interval will be about 12 minutes. Using a 50 kilohm potentiometer for R2 will give a maximum pump run time of about 35 seconds.
C1, the 1000 uF capacitor, will probably need to be an aluminium electrolytic. These are not very accurate, and their values vary somewhat with temperature and age. Here's one from Nichicon, one of several high-quality Japanese brands:
http://www.digikey.com/product-detail/en/UBT1E102MHD1TO/493-4500-1-ND/2650340
Here are some more links to Digikey part pages for you to look at:
1M potentiometer, slotted shaft:
http://www.digikey.com/product-detail/en/51AAD-B24-A25L/51AAD-B24-A25L-ND/3828851
1M potentiometer, four-turn (easier to adjust accurately):
http://www.digikey.com/product-detail/en/3856A-282-105AL/3856A-282-105AL-ND/3828850
50k potentiometer:
http://www.digikey.com/product-detail/en/93R1A-R22-A18L/93R1A-R22-A18L-ND/2564723
There are many possibilities.You can check out other potentiometer options using Digikey's selection table:
http://www.digikey.com/product-sear...sistors/rotary-potentiometers-linear/?stock=1
The only really important parameters are the resistance, and the mechanical style (mounting style, shaft style). I would avoid "conductive plastic" and wirewound ones; carbon is OK and cermet is the best quality material.
Digikey have potentiometers with values of 2.5 megohms and 5 megohms; you could use one of these, to allow you to use a lower-value capacitor. Reduce the capacitor value from 1000 uF by the same factor that you increase the potentiometer resistance by. You'll also need to increase the 50k potentiometer in line with the 1M potentiometer.
TLC555 (improved version of the 555 timer):
http://www.digikey.com/product-detail/en/TLC555CP/296-1857-5-ND/277502
NTD2955 MOSFET:
http://www.digikey.com/product-detail/en/NTD2955-1G/NTD2955-1GOS-ND/1484753
That MOSFET will be suitable for your application. Don't worry that it's rated for 12A; it's the cheapest suitable MOSFET on the list!
Here's how to connect it.
Connect the MOSFET's source to the +12V supply, which also powers the TLC555.
Connect the MOSFET's gate through a 22 ohm resistor to pin 3 of the TLC555.
Connect the MOSFET's drain to the positive terminal of the pump.
Connect the negative return from the pump to the circuit's negative rail (called the "0V rail").
Connect a diode across the pump, i.e. from the MOSFET's drain to the circuit's 0V rail, with its cathode (the stripe end) to the MOSFET's drain. This protects the MOSFET against inductive kickback (aka back-EMF) from the pump, which could damage it. A suitable diode would be the 1N4001:
http://www.digikey.com/product-detail/en/1N4001/1N4001FSCT-ND/1532742
You'll need to build the circuit up on stripboard, also called veroboard. Google stripboard assembly tutorial for information.
Any other questions please feel free to ask.