Respect in Saints Row 2 can be obtained in many different ways and determines how many main story missions you can do.
Respect Levels
Respect is shown as the purple bar around the bottom half of the status circle in the top right. Each time this fills up you get one added to your x?? value displayed on the bar. The number that is displayed determines how many main story missions (including Strongholds) that you can do before you need to gain more respect. Reaching a high enough x?? value and Style Rank (outlined below) will turn the bar to displaying an infinity sign, allowing you to do as many main missions as you like. As the x?? number increases, more Respect is required to fill the bar and add to that value.
Gaining Respect
Virtually anything remotely stunt-like or combat-related, as well as Activities and Diversions, will give you a certain amount of Respect. Combat tricks and driving tricks give the least, such as Gang Kills, Beatdowns, Big Air, Oncoming Lane, Two Wheels etc. etc. - usually in the order of 10-500 respect. Diversions come next, with anything from 300 to 1000 respect. Finally, Activities, and in particular the higher levels of Activities, will give the most respect, up to around 10,000-15,000. At low x?? values it may take around 7,000 to fill the bar, whereas at higher levels it can be up to 40,000.
Style Ranks
Style Ranks are obtained by getting a certain amount of Style Points. Style Points are gained in various ways, from buying new clothes, tattoos and piercings, to vehicles and boats, and even modifying cars at the Mechanic. Also, upgrading Hideouts will also confer Style Points. Usually the amount of Style Points you gain is proportional to the amount of money you spend on said item; lots of money = lots of style. Your current Style Rank and progression towards the next is shown by the small Saints symbol, number, and bar that appears below your health/weapon indicator in the top left. The number represents your current style rank, and the yellow bar represents your progression towards the next; once the entire ring is completed, you gain a rank. Each rank confers a 5% bonus to respect gained from Activities, so Rank 2 gives 10% extra, Rank 8 gives 40% extra, and so on. For the later levels of Activities, Style bonuses can give you as much as an additional 5,000 respect.