Do I need to watch the James Bond movies starring Daniel Craig (Casino Royale and Quantum of Solace) before watching Skyfall to understand the plot and characters?
Yes... and no.
Contrary to popular belief, and to every other Bond movie ever produced, it is entirely necessary to possess foreknowledge of the previous Daniel Craig Bond movies to garner the appropriate amount of appreciation for what is occurring.
The Daniel Craig Bond movies represent a canonical reboot, in which during the first film Bond earns his 00 status - a part of his history every single other pre-Craig film has been chronologically set beyond.
The Craig films aren't a prequel, they are a total reboot. The first two films (Casino Royal and Quantum of Solace) make a bigger proposition of linking the narratives together, but Skyfall is a direct continuation from this film.
On the other hand...
The diegesis of Skyfall is relatively self contained, and isn't directly dependent on individual plot elements from the previous films. There are no recurring characters that add momentum to the plot, and (as with previous Bond installments) no need for direct knowledge. The experience gained from the first two films is hugely advantageous in understanding the character's development, motivation and of course...
why he's over emotional about the death of his boss, who has been an advocate and mentor in the past two films
to put it on the Lucas scale: the modern James Bond films are closer to Indiana Jones than Star Wars... Indiana Jones is a character central franchise, in which each installment contributes to the overall construction and understanding of a specific individual, but each iteration is not totally dependent on foreknowledge of the last...
No, it is not necessary to watch the modern bond movies in sequence, nor is it necessary to watch any of the James Bond films in any particular order. A previous bond film was only referenced by a later film once in the entire series, and each movie can essentially be considered isolated from the others.
No it is not necessary for any Bond film. Each can be enjoyed on its own.
There are some generic aspects ('Q', 'M', innuendo, etc.) that one becomes accustomed to, but prior knowledge is unnecessary.
While I enjoy the light hearted approach of the early films, the Craig generation are closer in spirit to the relative darkness of the Bond character in the Fleming series.
I know I'm late but to set the record strait Daniel's bond is not stand alone. Notice how he is, when he first gets his 00 status he's wreckless and makes plenty mistakes he also learns his lessons that make him the man he is in Casino Royale and Quantum of Solace. And in Skyfall he has moved on and is a more experienced James Bond them he looses female M and meets Moneypenny and male M, therefore setting him up as the original James Bond we first saw with Connery. His Bond films are connected as you can see in Spectre also with the return of Mister White saying "I always knew death would wear a familiar face". This is referencing back to when Bond destroyed Quantum but White got away and now Bond will see the true cause of all his pain.