There are two general approaches to run flat tyres.
1. The Reinforced Tyre sidewall: this approach involves the manufcatures using a slightly different rubber compound to reinforce the tyre's sidewall, allowing it to effectively support the weight of the vehicle, even after a puncture or loss of pressure. The tyre will allow the vehicle to still function and handle as it would normally with acceleration and braking unaffected (you should check the manufacture guidelines for each of their run flat products before use).
The reinforced sidewall approach is more common than the second - the rubber clip approach - and an be fitted to any rim. Continental, Goodyear, Pirelli, Dunlop and Birdgestone all use this system.
2. The second approach is the rubber clip to your rim, used most commonly, and developed by Michelin. This is known as the PAX system. Unlike the 1st system it does not fit on to any wheel and requires specially modified rims in order to be fitted. There is a inner support ring fitted to the rim of the tyre, allowing you to continue driving (at limited speeds) on a tyre despite a puncture.
Despite being far less common, Michelin users have reported increased comfort levels compared to the reinforced sidewall approach due to the less ridged nature of the tyre.