Since this is very genre-dependant, I will just talk about platform games. Some games did that right, and some did not. The older Mario games as well as Donkey Kong Country allowed saving only at certain points in the game, which means you have to beat 3 to 4 levels in a row.
When you lose a life, you restart the level from the beginning or the last checkpoint you reached. When you run out of lives, the game goes back to the main screen and you have to start from the last saving point. That almost never happened in some games, since it was fairly easy to rack up 50 or more lives.
I never liked this system, mainly because you are not allowed to play for just 10 minutes and turn off the console. You just have to play until you reach the save point. Donkey Kong Country Returns on Wii autosaves after every stage, same goes for New Super Mario Bros Wii. That's way better, but the lives in both those games lose every purpose - you can easily get 99 lives in DKCR even before you beat the second world. And if you somehow manage to go "game over", you just lose your in-level checkpoint anyway.
The best system I can come up with is having no lives at all, unlimited continues and autosaving after every stage. The difficulty should come from within the stage and not from re-playing the same part over and over again because you're not allowed to save.
Fledermann
I respectfully disagree. While I agree that it's usually much better to be able to play a game more casually, not all games fit into that category. Part of the fun of beating a game like Super Mario Bros. was being able to do it without using continues. Yes, it took longer than 10 minutes to beat, but at the same time, the more you played it, the more familiar you became with the levels, the quicker it went by. It isn't an overly long game, and it does have shortcuts.
As for best system, there is no best system. Your system is good for some games but not for others. For instance, it'd be perfect for the Commander Keen series, but it wouldn't work as well for, say, Donkey Kong Country Returns for Wii, since dying there doesn't necessarily mean restarting a level. Those games are meant to be played with multiple people with the lives shared. If you used your system, that wouldn't be possible. That's just one example.
In short, it really depends on the genre and the target audience. There are inherently bad systems, but no saves and no continues is not one of them (though it's a much easier system to abuse). It works perfectly in arcade games, for instance, and games where the main objective is to get a high score.