My lingo is a little bit rusty, but I believe the term they're talking about is dodge-impact. Basically, if you press the Z, L, or R keys (the default shield buttons -- if you're not using the default scheme, just think "shield button") just as you hit a surface your character will not bounce off, but instead dodge into the ground or wall, instantly killing all momentum.
I'm going to go double-check this -- I'm not sure dodge-impact is the correct term.
Edit: Nope! Turns out I was 100% wrong. The above strategy might help you, but it's not the DIRECTIONAL INFLUENCE your friends were talking about. Here's a link that should clear things up: http://super-smash-bros.wikia.com/wiki/Di