I think /u/splurgingspurgeon answered very well but I'll add that players we, as a fanbase, really love are:
Zach Parise - alternate captain
Ryan Suter - alternate captain, one of the best defensemen in the league
Mikko Koivu - our captain
Devan Dubnyk - current goaltender. Was brought in to try and fix our goalie situation, and has been AMAZING. You will often hear what sound like "Boos" when he makes an amazing play. But it's people yelling "DOOOOOOOOOOOOOOOB!" It's so pervasive, that you hear the chants even in other teams' stadiums. This is why you see the picture "The Dube Abides" to the right. He saved our season from being a disaster and has catapulted us to the best team in the league since January. Nino Niedereiter - A player we picked up who his team had given up on and who has turned into an AMAZING star for us.
Jason Pominville and Thomas Vanek - They were star players for the Buffalo Sabres. We snagged them both, to Buffalo's sorrow and our joy. Vanek has a history in the Midwest (he is from Austria, but went to high school in Sioux Falls, SD where he led the Stampede to championships, then went to the University of Minnesota where he led them to championships, spent years with Buffalo in the NHL, and is now with us!). Jason Pominville was Buffalo's captain, now he's one of our best players.
Schroeder - Young player still in our system who sometimes plays for our minor league team, but has been phenomenal when called up to play for the Wild. Lots of us wish he played more in the big league.
Jason Zucker - Won us our first playoff game in years with this goal. He has launched himself into being one of our best players, sometimes single-handedly carrying us in games. He's injured right now but will hopefully be back in the next couple weeks.
Mikael Granlund - Has his ups and downs, but last year he won us repeated playoff games with heroics such as this overtime goal which won us our first game of that 7 game series. And then, the very next game, he blocked shot after shot after shot to win us the second of the series, even when he didn't have a stick. (Blocking shots hurts like fuck, by the way, but he held on to win us that game). As you can see, the Avalanche were on a power play (so we only had 4 players on the ice) and then they pulled their goaltender, which let them have 6 players against our 4. We went on to win that series, our first playoff series won in a decade.

The Avalanche Situation - Last season we won our first playoff 7-game series in a decade. We played the Colorado Avalanche. We went into this the lowest-ranked team in the playoffs, they were the highest-ranked. Multiple games went to overtime. It went to overtime of the 7th game, which we won 5-4 as you can see in this video. The rivalry from that series has been HUGE. I, personally, love the rivalry, but there is a LOT of bad blood between us and the Colorado Avalanche and it only seems to get worse every time we play. This year we've dominated them, but fans get into fights in the stadium, people get banned on reddit, it gets UGLY. This is going to be an on-going thing.

Zach and Ryan situation and why they're both alternate captains - Zach and Ryan played for different teams. Zach Parise was the captain of the New Jersey Devils and nearly led them to a Stanley Cup (they were 2 games away from winning it). Ryan Suter is one of the best defensemen in the league and was playing for the Nashville Predators. They both went on the market one summer. Everyone assumed they'd both stay where they were (Zach had nearly won a Stanley Cup, was a captain, Ryan was paired with Shea Weber who is arguably the best D-man in the league). Then, stunning everyone, they both signed with us, the Minnesota Wild. They are both from around here, Zach's dad used to play for the North Stars (our former NHL team), Ryan's dad played for the Miracle on Ice team. They both wanted to come home. They're also good friends, having played together for years and winning the USA's first gold medal in juniors. So they talked together and we got them both. As a result, Devils' fans are still sad (we don't blame them ) and Predators fans boo Ryan EVERY TIME he touches the puck (we are only starting to not be salty about this). So there's that!

We have a lot of Finnish players on our team (as in, from Finland) and you'll see a lot of love for Finland as a result (Finnesota , Finns = Wins, etc.).