A guy by the name of intheballpark wrote up a quick history for each team over on Reddit:
Mercedes has almost no common history with the all-conquering Mercedes team of the 30s and 50s, except that it's owned by the same company (Mercedes-Benz). The current team started life in 1999 as British American Racing (BAR), which was essentially a vanity project set up by Jacques Villeneuve and his manager Craig Pollock, who persuaded British American Tobacco to sponsor a team, with cars designed by Reynard. BAR were massively unsuccessful until Prodrive's David Richards took over in early 2002, who did a deal with Honda (engine supplier since 2000) to be Honda's main team from 2003. In 2004 Richards' leadership got the team up to 2nd in the championship behind the all-conquering Ferrari, albeit with no wins. Richards stood down at the end of 2004, and in 2005 BAR had a bad season, being banned for 2 races for cheating at the San Marino Grand Prix (deliberately running the cars underweight).
Honda bought the team for 2006, and although they won a race that year, their performance tailed off so much that by 2008 they were back of the grid, and Ross Brawn was brought in early in 2008 to improve performance. Late in 2008 however, it was announced that Honda was pulling out of F1 completely, and hurried negotiations saw Brawn buy the team himself and rename it Brawn GP, using Mercedes engines, and with Honda bankrolling the 2009 season. After a quick re-design of the rear end to accommodate the Mercedes engine, Brawn went to the first race and proved how much they'd turned the team around with a 1-2 finish. Brawn went on to win the championship, before selling to Mercedes for the 2010 season, having also persuaded Schumacher to come out of retirement. Unfortunately for them the 2010 car wasn't as good, and Mercedes haven't won a race since returning.
Another poster named shawa666 came back with:
Actually, The Current Mercedes outfit can trace it's roots back to a small organisation called Tyrrell. Which was bought by Craig Pollock to becom BAR.
To which intheballpark replied:
Actually no. What Craig Pollock did was buy the entry rights of the Tyrrell team, and disposed of the entire rest of it. Therefore I considered it to be a new team, as it consisted of absolutely nothing from the old Tyrrell team.
I still sometimes see members of this board treating BAR/Honda/Brawn/Mercedes as rebadged Tyrrell's, despite others pointing out that nothing came from Tyrrell except for a few sheets of paperwork. I thought this guy did a nice write up to clarify the team's history. The team we root for today really got its start as BAR.