Millennium was created by Chris Carter, the man who created the X-Files, and both shows shared some creative personnel. On screen connections were minimal. The only explicit connection was the character of Jose Chung who appeared in the Millennium episode "Jose Chung's The Doomsday Defense" and the X-files episode "Jose Chung's From Outer Space." David Duchovny, who plays Fox Mulder on The X-files has a cameo in "Jose Chung's The Doomsday Defense", but not as Fox Mulder. After Millennium was cancelled Lance Henriksen appeared on the X-files as Frank Black in the episode 'Millennium' in season 7.
Another possible connection concerns Albert Hosteen, a character seen in several episodes of The X-Files. The actor portraying him, Floyd 'Red Crow' Westerman, is featured in 'A single blade of grass' from the second season of Millennium, and though his character name is not mentioned in that episode, his character is very similar to Hosteen as portrayed in The X-Files (and like Hosteen, of Navajo descent). This connection is only speculation, but it's not unlikely the old Native American in 'A single blade of grass' was at least meant to be Albert Hosteen.
There's also a moment in the series which weakens the plausibility of Millennium and The X-Files existing in the same universe. In the episode 'Human Essence' (season 3), an episode of The X-Files (season 5's 'Kill Switch') is heard playing on a TV (heard, mind you, not seen). This would mean that The X-Files is as fictional in the world of Millennium as it is in actuality. It's probably only meant as a gag by the producers of Millennium, but it muddles things up a bit.
One direct connection, albeit a brief one, occurs in the season one episode "Lamentation". At around the 08:45 mark, when Frank is ascending the stairs at the FBI, you can clearly see Fox Mulder and Dana Scully descending the stairs in the background.
Another joke is played on the audience when Peter Watts (Terry O'Quinn) mentions the popular belief of humans being abducted and experimented on by aliens in concordance with the government, is 'Owls' (season 2). This is of course exactly what is going on in The X-Files, and thus also in the realm of Millennium. Watts feels such beliefs are ludicrous, which is interesting coming from a character who's played by an actor who guest-starred no less than three times in three different roles in the X-Files universe!
All in all, except for the 'Millennium' episode in season 7 of The X-Files, the relationship between both shows mostly consists of brief references for intertextuality's sake. It's never really mentioned to be taken seriously, they're just fun in-jokes for loyal viewers of both Chris Carter shows.