prefix-list is used generally when you want to control the routes(prefixes) that is being sent or received to neighbors in routing protocols like RIP,EIGRP,BGP, it can also be .for route tagging etc.
In prefix list for example
a.b.c.d/x ge y le z
x bits should always match in prefix while the subnets should be <= than (le) z and >=(ge) y
10.1.0.0/16 le 24 ge 16 will have 10.1.0.0/18 but no 10.1.1.127/25.
Distribute-list to used to suppress the routes either in inbound or outbound direction. Say from other end of eigrp neighbor router you are receiving x,y,z routes, but you want only x in your RIB. So you can deny route y and z. Similarly your router is advertising routes to its neighbor and you want to hide some routes from them, you can do so with help of distribute-list.
Gateway keyword is used to specify the neighbor from/to you are denying/sending routes. In simple term you have two RIP neighbors you want to send route only to one particular neighbor but not other one.
