There are quite a few themes that run throughout the series, including:
- the importance of family
- power and its corrupting influence
- envy and regret
Ultimately, you are asking a massive question, so I'll try and explain my interpretations of the themes one by one. Before I do, I'm assuming you've watched this whole show and thus this answer will be rife with spoilers.
Finally, this will be rather long...
The Importance of Family
Skyler and the kids
This is the obvious theme to start with as it is what starts Walt off on his entire adventure. Throughout the first series in particular, everything Walt does is aimed at providing for his family. He desperately wants to protect them and ensure that they have enough money after his death - he even goes as far as to calculate the exact amount he needs to earn, before he dies/retires from the business.
As the series goes on this theme retains its significance because Walt continually forgets it, instead acting for his own selfish benefit, a fact he finally admits in the final episode in the series, when he tells Skyler all his actions have been for himself.
The importance of family can also be seen in Walt's actions not just to provide for his family, but also to protect them. When he fears at different times that they are in danger, he immediately rushes to protect them.
Of course, Jessie is part of Walt's family too. During the series, Walt wants to keep him under control - but that doesn't mean he doesn't act to protect or provide for Jessie at times. He makes a point of giving him cash and looking after him following Hank's brutal beating of him.
Obviously, their relationship is complicated and Walt frequently treats Jessie in a deplorable way. But, there are many times in the series when he could have cut Jessie out of the business, and doesn't.
In addition to Walt's feelings for Jessie, we Jessie's feelings for Jane Margolis (his lover from Series 2) and the Cantillos.
He falls in love with both of them and feels responsible for them. In Jane's case, he is careless and selfish, leading her away from the "good" path and down into drugs again. Following her death, life appears to lose meaning to him, as he feels so guilty. With the Cantillos, Jessie desperately tries to provide and help them, even going so far as to give them a huge amount of money.
When Brock is poisoned, Jessie is outraged. When he discovers that Walt was behind it, he goes so far as to try and burn his house down, showing his overall family loyalty is stronger to the Cantillos than it is/was to Walt.
Hank and Marie
Despite being chased by Hank, Walt tries to protect him (at times). For example, in the Final Season, when Walt is at his worst and Hank is dying on the ground, Walt begs for him (unsuccessfully) to be kept alive.
Hank and Marie both show their loyalty to the White family by trying to protect (and even kidnap) the kids in the Final Season, to get them away from Walt and Skyler's corrupting influences.
There are other demonstrations of family importance in the series, such as the Twins coming to New Mexico following Tuco's death, or Hector Salamanca's loyalty to his family (leading to his death and murder of Gus).
In some ways, sin in the series is brutally and explicitly discussed, whilst in others it is completely absent.
In the series, Walt commits the crime of cooking and distributing meth. However, we never see any sign whatsoever that he has considered the effects of selling meth has on people. He is more concerned with protecting his family, making money and providing high quality meth as it is the staple of Heisenberg. He never appears to question the damage such an addictive drug does to the hundreds and thousands of people that will take it.
However, other sins are extremely prevalent in the series. We see Jessie battle with it continually, as he realises his own actions have hurt so many others (leading him to throw all his money on to the street as he loses interest in protecting himself any more).
We can also see varying levels of sin, depending on your definition of what sin is. We see Walt obviously selling meth and committing murders. Hank on the other hand works as a respected law enforcement agent, but is cruel and unforgiving to Marie during his rehabilitation. Skyler smokes, drinks and even threatens suicide whilst pregnant. Marie becomes a kleptomaniac, stealing from stores and eventually house viewings. Gus leads a criminal empire, dispersing his drugs across the South West of the country and murdering any who stand in his way (including the entire leadership of the cartel). Gale manufactures meth for Gus, which will lead to the addiction of hundreds and thousands of people - but has no interest in murder or hurting anyone directly.
Thus, we can see that almost every major character has committed some sort of sin. However, each sin is of a different importance. Almost all the sins we see committed have a reason. For example, Mike commits all his actions to protect his grand-daughter. However, it is left to the viewer to decide whether the sins committed by characters, compared to their living circumstances, absolve them or free them from judgement.
For example, in the Final Season, many fans were obviously happy to see Jessie get out. But lets not forget, this was a man who directly led Jane Margolis down a path of death, who murdered Gale, who manipulated an entire rehab group by introducing meth into them. However, as his conditions throughout Season 5 were so brutal, it was hard not to set aside those sins and allow him his redemption.
Ultimately, sin is around every corner in Breaking Bad.
Power and its corrupting influence
As mentioned earlier, Walt only admits in the final episode that all his actions have been for his own benefit. However, its been obvious to the viewers for a long time. Whilst there is some evidence of his corrupting influence on other characters, by far and away it is Walt's journey that is of significance.
As he grows his meth business, he becomes more irate at the idea of anyone having any control over it. He refers to it as his empire. When Hank states his belief that Gale was Heisenberg, Walt reacts with quiet fury, disgusted that someone else get credit for his brilliant work.
The power he derives, not just from cooking, but from cooking so well, leads him to be arrogant. However, this arrogance is only fed by the fact every character he meets in the drug world goes along with it, as they realise just how good he is. All of this contributes to his total corruption.
Envy and regret
These are probably my favourite themes of the series, as they are introduced so early and are so relevant throughout.
In the very first season, we see Gretchen Schwartz offer Walt money. He kindly refuses, but she discovers he has told his family she is providing the money. When she confronts him, we see the first taste of just how dark and twisted Walt is. He is bitterly furious about both Gretchen and Elliott and feels they have ruined his life by cheating him out of the company he helped found.
We never really find out exactly what transpired between Walt and Gretchen, other than the fact he left her rather abruptly, which led to his leaving the company and her marrying Elliot. This regret (to leave the company) never leaves him.
He reveals this to Jessie later in the Series, when he discusses how he was bought out for $5,000. Now the company is worth over $2 billion. He discusses how he checks the company's worth every single week. This self-inflicted torture sustains his envy and regret and makes him more twisted and bitter than ever.
Whilst the meth business gives Walt an outlet for this anger, it has clearly been underneath the surface for a very large time.
We see this envy and regret one last time, in the final episode of the series. In a tragic scene, Walt shows up at the Schwartz' house and manipulates them into leaving all of his money to Walt Jr. when he is old enough. He leaves $9.72 million - a huge amount, yet so pitiful compared to what he could have had. Would that amount even cover the house he is standing in whilst making the deal?
It clearly shows that whilst, as the old adage says, the drug business is easy money, it still is nowhere near the amount he could have had had he stayed with Grey Matter. Even at the end, his regret is still there.
There are of course countless other themes in the series, but these are, in my opinion, the four most important.
This truly is a colossal sized question, so I figured I had to stop here or write endlessly! Hopefully this will give some indication as to the major themes and any pesky friends with pesky questions in the future can be answered!