Above are the standings in the American League of teams that still have a playoff hope. The top three are the division leaders, with only Boston having clinched a playoff spot (although not necessarily the division, but that could come tonight with a Rays loss or a Red Sox win).
If you look at the last 10, 20, and 30 games columns in the table above you'll see that the Orioles are the very definition of a .500 ball club. They had a pretty good week, going 4-2 with series wins against both the Blue Jays and the Red Sox, but the teams ahead of them haven't really been cooperating.
This weekend the Orioles play four games against the Tampa Bay Rays, who are currently tied for the WC lead with Texas. If the O's want to leave Tampa Bay with a prayer of the playoffs, they need to win at least three of these games. Going 4-0 will give them a two game lead on the Rays and going 3-1 will result in a tie record. These are the only two acceptable outcomes.
Tampa Bay Rays
The Rays had a 4-3 week, winning two of three from the Twins and going 2-2 against the Rangers, their current competition for the top wild card spot. They started their series with the Rangers tied with them for the first wild card spot, and four games later that's where they remain.
The Rangers spent last weekend getting swept by the Oakland Athletics and losing all hope for the A.L. West division title. They couldn't put away the Rays in their four-game series with them, and now they'll spend the weekend playing the Kansas City Royals. KC has a slim playoff hope remaining and are just one game behind the Orioles, but we need to root them on to victory over Texas.
The Indians continue their cake walk to the end of the season. They swept the White Sox last weekend but couldn't move ahead in the WC race during the week thanks to going 1-2 against the Royals. They remain 1/2 game behind both the Rays and Ranger and with a weekend series against the Astros they could easily start next week with one of the wild card spots.
Kansas City Royals
As I said, they have a slight chance to make the playoffs. They're three games behind the wild card leaders and play the Rangers three times this weekend at home. A sweep could put them right up near the top depending on what the Rays and Indians do. The Royals could easily pass the Orioles, but that can't concern us right now. They need to beat the Rangers. And even if they do pass the O's in the standings, who would you rather grab that last WC spot, the Rangers or the Royals?
New York Yankees
The Yankees are 3 1/2 out of the wild card and have four teams to pass, so their chances at the wild card are slim to none. They do, however, play the Rays next week so if the Orioles sweep the Rays and the Yankees sweep the Giants this weekend, next week could be interesting for them.
TL; DR - Who should I root for this weekend?
You should root for the Orioles, Royals, Astros, and Giants.