I'm not going to go with my normal review format for a reason I will make clear shortly. I'll come out and say it straight from the start: Cowboy Bebop is my favorite anime series of all time. Cowboy Bebop is the greatest anime series of all time. No anime series will ever top Cowboy Bebop (in my mind). Since simply naming it's strong points would be useless since I would just name every category, I'll just highlight why it is so strong in some of those categories.
Characters: The crew of bounty-hunters on the ship Bebop consists of 4 members (and a dog). They all get a couple of episodes that heavily develop their characters and offer us a glimpse into their past and into their soul. Each of these characters (maybe not Ed) could easily star in an entertaining series revolving around their pre-bebop lives. That's how interesting they are. Amazingly, the same can be said for the characters that only appear in the single episode stories. The series as a whole easily has 30 rainmaker characters. Imagine if every character in a Gundam series were unique, likeable and interesting. Wouldn't that make a huge difference.
The main protagonist, Spike Spiegel, is developed a bit differently. Even though he gets a few more episodes revolving around himself and the characters from his past than the others, information about his past is given in snippets. We are teased with hints and implications and some answers are never fully given, leaving you to fill in the blanks. This only adds to Cowboy Bebop's charm, giving it a Casablanca-like quality in the way Spike, Vicious and Julia can refer and act in accordance with a complicated and heavy past without letting the viewers in on all the details. Not surprisingly, these episodes are also the best and most dramatic episodes of the series.
Music: From the first episode, the quality of the selection of music peppered throughout the drama hits you like a ton of bricks. Yoko Kanno, already an established soundtrack veteran from series like Macross and Escaflowne, supplies Bebop with her best work yet. Though jazz and blues-heavy, the soundtrack runs the gamut from heavy-metal to bubblegum pop. Great collaborations from artists like Mai Yamane(vocals on the OP-ED) contribute to making the anime pinnacle soundtrack. It's the only anime OST that I listen to a regular basis. More important than the quality of the music however, is its appropriateness. The OST works so well with the subject matter and is used so selectively well than you can feel it enhance the drama, and vice-versa. I think that effect is the true measure of its success.
Style: Not usually a criteria in a review, style is the most unique strength of Cowboy Bebop, and most difficult to explain :( More than the obvious cool and calm way most of the main characters act : their witty replies, their clutchness when the pressure is on, and their overall demure; Bebop is crammed full of pop-culture references and influences. You can see Bruce Lee/kung fu influence in the fight choreography, the 40's drama influence in the character interactions and conversations, the sci-fi influence in the dramatic space-action scenes, on top of all the music and writing influences sprinkled and reflected over and over again throughout the series. When I'm debating the virtues of Cowboy Bebop with someone, one of the arguments I like to use to describe it's success is the quality of the ending scene of every episode. The last 30 seconds or so of every episodes is almost always a memorable and fitting ending to the story. It can be a dramatic pause to a contemplating character, or an ironic and clever turn of fortune that leaves its victim bemused but that final scene bring the whole episode together perfectly. Though difficult to explain in words, the reaction is quite palpable when witnessing it for the first (or second, or third) time.
Every 6 months or so I pick up my Cowboy Bebop DVDs and watch all the episodes over a period of a couple of days. Whenever I'm done, I never fail to have the urge to call up a friend or shoot them an email to pressure them into watching Cowboy Bebop, even to those friends with zero interest in anime (weirdos!). No other anime series has had such an effect on me and I know none other ever will.