The writers have a few ways of getting story to you: assignment, request, and bumping into it.
The first way of getting story is by assignment. The big honcho of the game gives you things to do. This is straight forward mission stuff. This is the stuff of the primary plot.
The next way is by request. You know people and crew members. They ask you to do something for them, either directly or indirectly. These are just as mission oriented, however, they usually act to elaborate and explore that particular character. When you've done that character's story mission, that character also gets a bump in power and a new outfit, so you get some perks to go with the story. I think that the crew-centric story hook is a good innovation. It serves multiple purposes well.
Finally, there are things that you just bump into. You meet someone on the street who asks a favor, or you detect a pirate base under you while you are exploring.
I guess that the fourth way of getting story is by involuntary cut scenes. For example, the into and the finale. However, they don't provide gameplay, not in a strict sense, so they act more as framing devices.
In ME1, you also got story when you entered certain systems for the first time. I haven't seen any of these system-triggered stories yet. I suspect that they proved irritating or annoying to some part of the playing base, or otherwise made players or devs unhappy.