How much grease accidentally going down a drain is a concern?
In general, a one-time slip-up dumping grease down the sink won't cause any clogs. Grease creates clogs over time, by building up layer upon layer inside the pipe until water cannot pass through (easily) anymore. 

This is most usually a problem with greases that are solid at room temperature but liquid at cooking temperature (or under hot water). When hot, it is liquid enough to be poured down a drain, but very soon after (i.e., while still in the pipes) it cools enough to solidify. It does this on the walls of your pipes. Cooling rates aren't uniform, so the grease doesn't always cool off and solidify into your pipes at the same place each time you pour it down the sink. Over time, enough eventually builds up to cause a blockage. Very similar to plaque building up on your artery walls and causing a heart attack.

Any kind of improper slope of your drain can exacerbate this, and cause the buildup to occur faster. A lot of bends in the pipes, excessively cold pipes, or debris in the pipes could accelerate this as well.

Running hot water down the drain will only push the problem further down the line, as you'll still have the problem of the grease (and water) cooling and solidifying.