The simplest way to increase catch rate is to achieve "Nice" or "Great" throws. To get those you have to land the Poke Ball inside the coloured circle. If the circle is just inside the outer grey circle, you'll get a Nice throw, which increases the catch rate by 1.15×. If it's around halfway you'll get a Great throw which increases by 1.5×. There is also an Excellent throw when the coloured circle is very small (1.85× multiplier), but that's difficult to achieve most of the time.
Second option is Razz Berries as Toucanadian said, which increases catch rate 1.5×. There is also a Golden Razz Berry which has a 2.5× multiplier.
Third option is to use a Great Ball or Ultra Ball, which increase the odds by 1.5× and 2× respectively. You get access to these as your level increases.
The final option is to use a curveball. This means holding and spinning the Pokeball in place, then flinging it to the side so it curves back to the middle. This increases the catch rate by 1.7×, however, it's pretty difficult to achieve reliably. Practice makes perfect though!
Of course, combining all the options above increases chances more and more. A curved Ultra Ball landing a "Great throw" would increase the catch rate by 5.1×!
