Decided to give it a stab. These are just what pop into my head. They aren't an objective view that takes all factors into account so arguing them with me will be pointless. In some cases they are strongly based on older history (Bears), in others entirely based on recent history that ignores past greatness (Browns), and still others a mix. Some are even impressions from growing up that have never left me and I'll always identify the team by (the fan support at Arrowhead that wowed me as a kid).
Cardinals: Consistently poorly run team
Falcons: Occasionally relevant but most always an also ran
Ravens: Defensive juggernauts, great management, but somewhat detestable ownership
Bills: Loveable losers you can't help but feel for
Panthers: Questionable support, a case of over-expansion?
Bears: Tough, black and blue, defense, running the ball.
Bengals: The team you keep expecting to break through into the elite and they never quite make it
Browns: Poorly run team, fans deserving of others pity for what they've been through.
Cowboys: Americas team but undeserving of the name. Overhyped.
Broncos: Most advantaged team in league (Mile High), seem to always have a high-passing QB. Quality club.
Lions: Mismanaged team, pity the fans.
Packers: Great history, loyal fanbase.
Texans: Consistently find ways to self-implode.
Colts: Successful on field history, but tarnished past to get to Indy
Jaguars: A case of NFL over expansion
Chiefs: The amazing football environment in Arrowhead.
Dolphins: Team with lost/lack of identity
Vikings: Never quite able to emerge from the shadow of rest of NFC North
Patriots: The either love 'em or hate 'em team.
Saints: Loveable losers
Giants: Tough defense, running game, cold weather team
Jets: Inept. Even their competence turns into inept competence.
Raiders: The NFL's black sheep
Eagles: Frequently relevant but always over-exposed
Steelers: Blue collar, defense and running. Great decision making focused on long-term
Chargers: The bottle rockets of the NFL. A glitter team that often looks great but ends up with only a tiny bang
49ers: History of champions and great decision making (3 years ago this would have been different)
Seahawks: LOUD. Underdogs with great fan support. LOUD. Did I mention LOUD?
Rams: The NFL version of Major League, minus the Georgia Frontiere stripper cardboard cutout
Buccaneers: Defensive minded, incompetent offenses
Titans: #### you Bud Adams
Redskins: Blue collar, winning in the trenches, history of excellence