What's the difference between day-to-day training and formal training? Not much, really, when you get right down to it.
Day-to-day training happens everytime Caysun and I are together. It happens when she's playing, when she's going to the bathroom, when she's wandering about the room. I call this kind of training "the rules of the road," because it's the way I teach the pup my expectations about how dogs behave.
It's as simple as moving my legs when she puts her paws on me. In just two days, Caysun's figured out that sitting elicits so much more attention than putting her paws on me. She's learning that leaning her front paws on the cabinets when I'm fixing her dinner isn't nearly as good as sitting. She's learning that there's a very specific path (and door) that leads to the outdoors where she goes to the bathroom.
Formal training will begin today. Formal training usually happens in a training session. I separate her from the other dogs, I have a specific goal and behavior in mind, and we work toward that goal, and mark our progress. Eye contact, for instance, is one of the behaviors I'll work on with her today. I will set out with the specific goal of clicking her each time her eyes meet mine.
Both types of training are equally important. You can't just do one of them and expect a well-behaved dog. Eventually, we'll work the formally trained behaviors into the day-to-day training, as Caysun becomes more fluent with the behaviors and is capable of doing the behaviors in an everyday environment (with the other dogs around, in the face of larger distractions, etc.).