The single most important requirement for defensive handgun ammunition is penetration. Without penetration, no bullet will do enough damage to stop an attacker. There are two camps; slow and heavy vs. fast and light (usually of the expanding variety), with compelling arguments on both sides.
Jeff Cooper used to say "Do not put your faith in expanding bullets". He was of the opinion that introducing another variable into the equation was unwise, and advocated .45 acp hardball for all your self defense requirements. There has been significant progress made in the design of hollowpoint bullets over the years, so his observation may not be as true as it once was.
However, the requirement that a bullet penetrate deeply enough to damage or destroy vital organs has not changed. The bullet must have mass and velocity sufficient overcome outer clothing, skin, flesh, fat, and possibly bone. Any 'major' caliber (9mm, 38, .357, .40, .41, .44 mag, 45acp, etc.,) will meet this criteria.
The next most important element is shot placement. It may do no good to hit an armed attacker in a non-vital area, even with a .44mag. So the next most important consideration relies on the shooter rather than on the bullet. Having a firearm with a capable round is only half the equation. Without the skill and mindset to use it, you're not prepared to defend yourself. Half measures availed us nothing. You need to be willing and able to place two shots in quick succession in an area no larger than your fist, under pressure in low light. That requires familiarity with your weapon and a lot of careful practice.