Martingale. Bet one unit on an even-money proposition. After a loss, double. After a win, revert to one unit. Each win recovers all prior losses in the sequence plus one unit of profit.
Updated statement of the mathematics. Martingale is often defended with the claim that "an infinite losing run is impossible". Stated precisely: the probability of an unbroken losing run tends towards zero as the run lengthens, but is never exactly zero, and the required stake grows exponentially. After n consecutive losses the next stake is 2ⁿ units and cumulative exposure is 2ⁿ⁺¹ − 1 units. From a $1 base, a run of ten losses requires a $1,024 eleventh bet and $2,047 of total exposure, and ten consecutive losses on a single-zero even-money bet happens roughly once every 700 sequences, which is not rare. Two hard walls end the system: the table maximum and your finite bankroll. Neither can be argued away.
Anti-Martingale (Paroli). The inverse: double after a win, reset after a loss. Caps downside to the base unit, but requires you to choose an arbitrary stopping point, and a single loss at the top of the ladder erases the run.
Fibonacci. Stake follows 1, 1, 2, 3, 5, 8, 13, 21 and so on. Move one step forward after a loss, two steps back after a win. Slower escalation than Martingale, therefore longer survival, and correspondingly slower recovery.
D'Alembert. Increase by one unit after a loss, decrease by one after a win. The gentlest common progression: smallest drawdowns, smallest recoveries.
Flat betting on outside bets. A constant stake on dozens, columns or even-money boxes. This is the sensible default for newcomers, because it maximises time-in-play per dollar, produces the flattest variance, and keeps your hourly expected loss predictable. Move to combination and inside bets only once you understand what the payout table is buying you.