The Break-Up
The breakup of the Beatles was not an easy one. As many know they did not part on good terms, especially John and Paul, and the whole process would drag through legal proceeding well into the seventies. The breakup was caused by a slew of different things. Contrary to much popular belief, it was NOT Yoko who broke up the Beatles. It is understandable why she was blamed and she was a contributing factor, but with a group as influential as the Beatles breaking up, it is not surprising people looked for a scapegoat. Yoko was the easiest one to blame, so she was it.
As I started with Yoko, I might as well elaborate on why she was a contributing factor. Yoko contributed to the breakup because of her  influence on John. Yoko influenced John with her avante garde art and he wished to reproduce more with her. She also changed him as a man, as many women do, because he knew she was the one for him. She became his first priority, putting the Beatles on the backburner. This is not to say her influence is negative. On the  contrary, the Beatles knew they would break up eventually and change styles and they only grew up and changed priorities. Linda was also blamed for the breakup but not to the same degree Yoko was. Paul, like John, became more fond of his wife than his group, although the Beatles were never on the backburner because of her. Paul was more balanced with his love and his job. Anyhow, the wives contributed to hte breakup but they received the brunt of the blame and shouldn't have.
Another factor was the introduction (or not) of Allen Klein as their manager. Apple, the Beatles company was losing money very fast, and Brian, before he died gave the Beatles a very bad contract. John decided he was going to have Allen manage his solo work. He had heard about Allen when he found he managed the Rolling Stones and they were making much more money than the Beatles. George and Ringo agreed with John and wanted Allen to manage the  group, but Paul was sternly against it. He preferred Lee Eastman to be their manager. Lee was Linda's father and the other Beatles objected to the obvious bias. Many arguments occurred over this and led to much  tension within the group. The Beatles did end up signing Allen, and as promised, he did make them much more money. However, in the mid 70s all the Beatles filed suit against him and received a settlement from him. John was most scathing towards Allen when he wrote "Steel and Glass" from his Walls and Bridges album.
With the death of Brian in 1967, the Beatles were left without a manager and for a while managed themselves. This allowed each one to do what he preferred, and many times they  really enjoyed that freedom. Brian was the glue that held the Beatles together, and with his passing there was nothing there to hold each one down to fit in  with the group.
I would say the main reason for the Beatles breakup is change. The whole time they were a group they evolved. Many people change from their teenage years until they are about 30 (let's hope most do), and the Beatles as human being did this. However, with the pressures of fame, especially the fame they experienced, changed them even more. The change is reflected in their music and one can see that by the time of Abbey Road each had his  own style of  writing and own style of songs. To be in the Beatles was stifling. John wanted to play straight rock and roll, Paul liked playing very melodic songs, George was into spiritual writings and  Ringo wrote whatever style he saw fit. These ideas clash obviously, but added to that is the  limited number of songs each could contribute to an  album. George  was especially stifled because he was limited to 2 songs an album, and he was as prolifec in song writing as John or Paul.
The Beatles were the greatest, most influential icons of the turbulent 60s (yet another factor). Their  breakup was felt by many  many people. Through out the whole  period of the 70s each was bombarded with the question of when the Beatles would rejoin. It probably would have continued into the 80s if John hadn't tragically been murdered. Each was very scathing and many times petty when it came to their disputes. It was hard to believe that the group that had sang "All You Need Is Love" was now exhibiting such violent behavior. Ringo was probably the most civil and that is probably why he is considered to be the one that would have been most likey to get the reunion. He often had the   others work on his albums.
The breakup of the Beatles is sad but it is something that was inevitable. It was a very stuffy situation and it is not surprising that each, at one time or another, quit. The saddest part is that they never got a chance to reunite, although the anthology comes close. Also it would have been nice if the disputes were not so petty and ruthless. The way they handled the breakup could have been much better, some of the things they said and did to eachother  are horrendous, and are no way how people should treat their friends of some 15 or so years.