Standard DVD players are 480i - always have been, always will be.
Progressive scan DVD players deinterlace the 480i to 480p. Many of the new $100+ players (and some below) perform upconversion to 720 or 1080i. This is just interpolation between the real pixels to "enhance" the picture, with varied results.
Telecine is not related to deinterlacing, although in many respects, the two often come together. Telecine allows film-based sources (typically movies, 24fps) to be faithfully reproduced on video displays (~60fps in US). The original source is encoded at 60 fields per second in such a way that a player or display that can recognize it can decode it and reconstruct the original 24 frames per second (reverse pulldown). Reverse pulldown is common on DVD players these days, and there are some HDTVs hitting the market that can pulldown a 1080i signal. This will be important for film-based HD source material in the future, but since most broadcast HDTV programming is video-based (60 fields per second), it's not so useful right now.
Finally, just to complicate matters, many fixed-pixel HDTVs have been accused of not doing a proper job of deinterlacing a 1080i source, basically displaying a line-double of a single field. Obviously, this might be slightly better than 480p (it's essentially 540p) but it does not reproduce the level of detail of real 1080i.