Do I have to have seen any other James Bond films to watch Spectre?

According to this interview with Sam Mendez (Spectre's Director) he certainly strived to make sure that the film works as a stand-alone piece, without the audience possessing substantial knowledge of the Bond series of films:

MENDES: I love that challenge; it’s a great privilege and excitement to express myself in that way to so many kids. I just hope that I don’t let them down and there are those pressures because at the end of the day, you may make the movie you want, but this is for the audience. And you do pray and hope that everyone believes in it as much as you do, including the kids, and you have to remind yourself every day you’re working on it that for many kids it will be their first Bond movie just as Live And Let Die was my first, and like From Russia With Love or Dr. No were for others. You hope it will make kids go seek out the earlier movies, which are now so accessible to them.

Obviously there are extensive references to earlier films and it can't possibly hurt to watch some of the previous films that have referenced SPECTRE (especially 'Dr. No', 'From Russia With Love', 'You Only Live Twice', 'On Her Majesty's Secret Service' and 'Diamonds Are Forever') but neither is it in any way necessary.

While the other answers are right that Bond films in general are usually entirely independent of each other and showing James Bond on a self-contained and independent adventure, there is a little more to it with regards to this specific film.

While the story of this film is also largely independent and can certainly be enjoyed and understood without having seen any previous films, it is definitely contributory to have at least seen the previous 3 films with Daniel Craig as Bond. Contrary to the majority of previous Bond films, the Craig series relies quite a bit more on cross-movie continuity, even if the films' main stories still stay self-contained. This was first noticable in Quantum of Solace, which was to quite some degree a direct continuation of Casino Royale. Then Skyfall again stayed quite independent. But with SPECTRE the filmmakers try to somewhat close the circle to the previous Craig films, relating them to the new film's main storyline. So while you will still get and appreciate the main storyline of SPECTRE, you might not immediately grasp the impact of the bigger picture that the filmmakers attempted here, nor recognize one or the other recurring character from the previous films.

In addition to that, the Craig series also relies on a much more coherent cross-movie character development of James Bond himself. And as in Skyfall, this movie yet again delves a little into Bond's past. And there's also other characters from not only the past Craig films but James Bond mythology in general whose knowledge might add to the appreciation of their reimagining in this film (or not ;-)). But those are more soft requirements.

So to sum up, you certainly don't have to have seen any previous Bond film to understand and enjoy SPECTRE, but it would add a great deal to getting its full impact if you have at least watched the previous films with Daniel Craig, primarily Casino Royale and Quantum of Solace (in that order!).

_{All I know is that 007 is some type of a code given to a secret agent with 001 being the leader.}

_{This is not entirely accurate, though. We never get to see many other 00 agents, let alone a 001, and there doesn't seem to be a hierarchy among them. What you really need to know is that James Bond is a British spy (working for MI6) and his boss is called M.}