Fantasy Football Definition of "Flex Player"
A "flex player" is also known as a "wild card player" in fantasy football. The flex player position is a spot in a team's starting lineup which allows the owner to choose a player from two or more different positions. This gives a fantasy football team flexibility in which player positions they select to start every week, allowing the team to select their strongest possible starting lineup. Flex positions have become much more common in the last 10 to 15 years of fantasy football.
Flex Player Strategies
Flex player positions not only gives teams flexibility when injuries or bye weeks happen, but it also allows a team to choose what their focus will be in the fantasy football draft. This introduces a degree of draft strategy which doesn't generally exist in leagues with rigid starting lineup positions. For instance, many old leagues would require a team to start 2 running backs and 2 wide receivers. In leagues with a flex player, you might be able to start 1 running back and 2 wide receivers, as well as a flex player who might be a running back or a wide receiver. Or you might be required to start 2 running backs and 2 receivers, then start a flex position player who can be either a runner or a receiver.
Offensive Flex Player
When teams fill out their weekly starting lineups, then, one might start a 3 RB/2 WR set, while the other starts a 2 RB/3 WR set. Each of these teams is presumably starting its strongest lineup, and is employing a different theory about which position will be more effective that week. In leagues with multiple flex players which also allow tight ends to start (and have scoring systems weighted to make tight ends more even with RB's and WR's), you might see a high amount of diversity. You might see one team with a 3 RB/2 WR/1 TE set, another team with a 1 RB/4 WR/1 TE set and another team with a 2 RB/2 WR/2 TE set. In these leagues, you'll see teams employ radically different fantasy draft strategies.
I've seen teams in the above "multiple flex player" formats draft 4 wide receivers in the first 5 rounds, believing that superstar receivers fall further in the draft and they get injured less often. (This I've seen in point-per-reception leagues. It would be insane in a league without that scoring rule.) In the same league, I've seen another team draft three running backs, trying to get as many RB's in their lineup as possible. I've seen a third team ravaged by injuries and end up starting two tight ends, because these were two of their most consistent positions. This makes for intriguing fantasy drafts, because they don't always follow the same straightforward RB-RB-WR philosophy that fantasy drafts followed for so many years.
Defensive Flex Player
In IDP leagues, you'll often see defensive flex players or defensive wild cards featured. In this system, you start individual defensive players from a variety of positions: usually defensive lineman, linebacker and defensive back. In these, you might be required to start 1 of each position and then have 1 defensive flex player. I've even seen leagues which require the use of three IDP defensive players, but allow them all to be flex players.
Whatever your IDP system, I would suggest you always start as many linebackers as possible. The way most fantasy football IDP scoring is set up, it allows points for sacks, interceptions, fumble recoveries, passes defended, solo tackles and assisted tackles. (Sometimes passes defended and assists aren't allowed.) In either case, players are going to pile up tackle stats much more consistently than sacks and turnovers, so your linebackers, who generally collect the most tackles, dominate the IDP formats.
Try to start a productive middle linebacker if possible, because MLB's usually lead their teams in tackles. On really bad teams, strong safeties occasionally post huge tackle stats, too, because their front seven let the running backs into the secondary, forcing safeties to make touchdown-saving tackles. So if you employ the 1 DL, 1 LB, 1 DB and 1 Defensive Flex Player format, I would suggest starting 1 defensive end, 2 linebackers (middle linebackers, most likely) and 1 strong safety. Ironically, you're probably better served getting most of these players off of losing teams, because these defenses are on the field for more plays on average, and therefore will have more chances to build their box score stats.