Icon -- Icons have been headliners for decades by this point, and have recognition outside of comic books due to enough exposure through movies and cartoons that a person who hasn't read comics for a number of years, or who has never read comics can name them. They are known by their civilian identities almost as well as their superhero identities. They often have multiple books, and a number of headliners as satellites. Superman, Batman, Wonder Woman, Spiderman, and Robin (Dick Grayson) are all Icons. There are varying degrees of Iconism (Hal Jordan's status is arguable, for example, as DC is making the push for Iconic status, and he has recognition for Superfriends, but John Stewart has more recent recognition for JLU. However, John Stewart is not a Headliner, like Hal is, and is rarely used as more than a supporting cast member in the comics) but it's safe to say that an Icon naturally leads a Franchise. A Franchise consists of more than one book, and some exposure outside of the insulated comics community in the form of merchandising or alternate media. An Icon is the sort of character that will be proclaimed the "The REAL ____" by people who have never actually read more than a couple isolated issues of a franchise (Kalinara has a story about this one for you).
Headliner -- Headliners come in two basic forms. Independent Heroes with their own ongoing solo books (such as Firestorm, Manhunter, Black Panther, Iron Man) who do not have Icon status, or Franchise members who are not dependant on an Icon's or Team's constant presence for the storylines (though sporadic guest appearances will occur). A Franchise will often have several Headliners who receive occasional guest help from their Icons. Supergirl, Catwoman, and Kyle Rayner are Headliner members of Franchises. They get guest appearances from the other Franchise members, but the Icon's presence is not in every issue, and their plots do not revolve around their relationship to that Icon (other than, he is the lead of the Franchise).
Team Player -- Members of a teambook who do not have solo ongoing series. If a member of a teambook has their own ongoing series, they are promoted to Headliners or Icons. Team Player is the status of a character who has no home other than the teambook itself (Stargirl, Power Girl), and guest appearances or supporting roles in other Headliners's books (Jay Garrick, Alan Scott). Team Players will often have their own miniseries once they gain enough popularity, or need a storyline away from the rest of a group, but their ultimate home is the teambook itself. The Team Player works in shared spotlight, though internal group dynamics do often make some characters outshine others (You have your Leader, your Peacemaker, your Anchor for the audience to play into, your Token Characters, and your Useless Romantic Soap Character, just to throw some examples out). Team Dynamics are a post for another day.
Backup Feature -- This is a now defunct status, as most comics no longer have backup stories, but some characters were created for solo stories in the backup features of Headliner and Icon books. They had their own plots and the stories were centered on them. They have pretty much all been demoted to Professional Guest Star or Heroic Support.
Professional Guest Star -- Usually a Headliner after her book's been cancelled, or a Team Player who is no longer on the team (or the Team Book is cancelled), but sometimes someone is created specifically for this role. This is a Hero, not a Civilian, who shows up to help the Headliner or the Team. They have no home, they are a nomadic people. They do have fans, and they do show up from time to time in various books. Sometimes this is just an excuse for a writer to use a character they like. Sometimes this leads to a promotion to Team Player or Headliner as it serves as buildup for the character's new role. Sometimes it leads to a demotion to Heroic or Civilian Support. Sometimes, they die.
Heroic Support -- The hero here is a regularly occurring member of the Headliner/Icon's cast, rather than a guest star. This is lower on the hierarchy than a Professional Guest Star because you have less independance. Heroic Support is completely subordinate to the Headliner. Their character development depends entirely on how it affects/reflects/leads into the Headliner's plot. Like a Professional Guest Star, Heroic Support is often a character rescued from Limbo (when their home book was cancelled), but sometimes they are a reformed bad guy or actually created for this purpose (Sidekicks?). Yes, a Heroic Support can become an Icon through exposure (Dick Grayson), but this leads to a promotion to Headliner. You can get a miniseries, but usually not an ongoing. Oftentimes they are captured/injured/in over their head and need to be rescued by the hero, who will solve the plot themselves. Sometimes they rescue the hero (this is usually so that the hero can come back stronger next plot, or so that the next storyline the hero can reflect on their weakness and relationship with the Heroic Support who rescued them). Black Canary (Green Lantern/Green Arrow, Green Arrow) Shade (Starman), Pied Piper (The Flash), Donna Troy (Green Lantern), and Jade (Green Lantern) have all been Heroic Support in the past. Todd Rice/Obsidian (Manhunter), Nemesis (Wonder Woman), John Stewart (Green Lantern), and Storm (Black Panther) are currently Heroic Support. Most Law Enforcement professionals count as Heroic Support, because if they get names and recurring roles they are usually badass enough to pull their weight (Jim Gordon, Cameron Chase, the O'Dares, Chyre and Morillo).
Note About Villains: Villains exist in their own hierarchy, though I would say that when a hero is turned into a villain, they would fall about here on the Good Guy Hierarchy of Superhero Comics, between Heroic and Civilian Support.
Civilian Support -- The lowest level of Character Status in Superhero Comics where you can actually have a name. Yes, some civilian supporting cast members are infinitely cool (Lois Lane, Alfred) but ultimately they are Civilian Support. It's a perfectly respectable position, for a civilian. For a hero, its the pits. And usually, when a hero loses their powers, this is where they end up (Mark Shaw in Manhunter, Mikaal Tomas in early Starman -- he was promoted to Heroic Support by the end) unless they have such impressive non-super abilities they can pull their weight like a member of the Batclan. While even Civilian Support can get a miniseries or even historically an ongoing (Lois Lane, Jimmy Olsen), their entire storyline revolves around the Icon to which they are attached and their relationship to the Icon is announced in the book title. Everything about Heroic Support applies, except you don't get a costume and you aren't expected to pull your weight, which makes it all the more impressive when you do get to do something.
Background Hero -- You get to fight in the background. Pretty much every unnamed Green Lantern is on this level. If they were ever named, they are Professional Guest Stars (Former Backup Features, msot likely), Heroic Support (Green Lantern and Ion), or Team Players (in Green Lantern Corps)
Background Civilian -- "Look! Up in the Sky!"