Basic (rechargeable) battery lessons:
Batteries create electricity by reacting chemicals together, on the one hand you have basic batteries which work on exactly the same principles as a galvanic cell:
http://chemistry.about.com/library/weekly/aa082003a.htm
Rechargeable batteries work on the same principle but instead of a one way reaction you use a reversible one (often a reaction with multiple stages with the intermediate stages being reversible). Applying electricity of sufficient voltage the other way reverses the reaction, naturally the chemistry is complex and fairly expensive (and possibly dangerous). Cheap makers almost invariably skimp on this and this is why people do not suggest cheap rechargeable batteries and alas with such a rep no good makers really venture into the market of proprietary batteries as few buy third party batteries at or around the price of official ones. Trouble is the voltage and size is an odd one and conversion is not exactly efficient.
Second to this is the problems associated with chemistry like self discharge and deep discharge (which can also be related).
Batteries produce electricity by oxidising things and most oxides are not know for their low resistance (as these oxides are internal to the battery such a phenomenon is known as internal resistance and is important in the next paragraph) so even without things connected to the terminals charge can be lost. Long story short some things fully oxidise and can no long be used for a reaction (lower storage) and deposition (blocking other particles from reacting; also lower storage and potentially higher resistance)*.
Deep discharge is not much of a problem on the DS as Voltage is related to apparent concentrations within a cell and this Voltage drops as the battery discharges for the reasons above and is more likely to become too low before deep discharge becomes a problem (this is also how the DS and over devices "know" the battery is getting low). Before someone mentions memory effect:
http://www.dansdata.com/gz011.htm
Internal resistance: now internal resistance means devices connected to a battery can only draw a limited amount of current (one reason why you may have to stick several batteries in a device) and a workaround for this is to stick several batteries in parallel as it will then cause the batteries to each provide a far lower current from each of them but ultimately a sufficient amount to power the device. While this is usually used for high power devices it is also useful for long term powering of devices (indeed I made one for an original DS).
Amp Hours is a measurement of how long a given battery can sustain a given current, it is used when you have a set Voltage (which you do here) as it makes the maths simpler than using power and/or energy equations. 1200mAh means if the battery is working properly a current on one milliamp can be sustained for 1200 hours (note 1mA is a tiny current), a draw of 2 millamps means 600 hours 3mA is 400 hours......
*You can reverse this in many ways including high current such a charged capacitor (remember discharge is exponential) or a suppy capable of delivering a large current (like a welder although a desk supply:
http://web2.murraystate.edu/andy.batts/ps/powersupply.htm can work for these smaller devices) or instead you can use high frequency pulse to perform a similar feat (note that your battery chemistry and the problems faced may not be all that receptive to this). Note these techniques may not be the safest of things to do.
More reading is you feel like it, it covers more things in depth and is probably worth reading from a design perspective:
http://www.powerstream.com/BatteryFAQ.html