Warning: Major spoilers up ahead.
Family and abandonment
Mawaru-Penguindrum has a theme of "family" and "abandonment" prevalent with each of the characters. Each major character in the show has experienced some form of abandonment and dealt with it in different ways, resulting in their current personalities.
The Takakuras are the central characters in the show, and as such, their family affects the entire plot. They have been abandoned by their parents, who are revealed to be part of the group which instigated the bombings which killed Momoka sixteen years ago. This abandonment resulted in the family thinking that the misfortune in their lives are a "punishment" bestowed upon them.
Individually, each of the Takakura siblings have experienced abandonment in their lives. In the final episode, it's revealed that Shoma and Kanba have been abandoned somehow, symbolized by their cages (whether this is symbolism or reality may be up to debate). Kanba was chosen by the fruit of fate (the apple), but decided to share it to Shoma so that neither of them will be abandoned.
Himari was initially set to be sent to the Child Broiler, a place for those who will "never amount to anything". Presumably this is because of her terminal illness. However, Shoma decides to share his part of the fruit of fate to Himari so that she will be saved.
Masako, as Kanba's sibling, was abandoned by Kanba and her father. Her family was also figuratively abandoned by her grandfather, who did not care about his children and believed that only those who had the willpower not to be "crushed" can survive. Masako channeled this into determination for bringing back Kanba into the family.
Tabuki was sent to the Child Broiler as well, but instead of someone sharing the fruit of fate with him, Momoka saved him from his fate.
Similarly, Yuri was saved by Momoka using the diary's cursed fire. Yuri was also figuratively abandoned by her father, who saw her as an ugly person and decided to "carve" beauty into her body.
Since both Yuri and Tabuki were saved by Momoka, they possessed a loving devotion towards her, which went wrong when Momoka died. They channeled this abandonment into revenge against the Takakuras. In the end, both of them only wanted someone to accept them as people, and they found this acceptance in each other. In a sense, they received the happiest ending among the cast.
Sanetoshi felt that he was abandoned by the world, having "escaped the box", as he put it. He felt that everyone else was trapped in their own boxes, happy and safe. Being someone who escaped from his box, he felt that everyone else should escape the box through his "Survival Strategy". He thought he found a kindred spirit with Momoka, but was again abandoned because Momoka didn't agree with how Sanetoshi wanted to go about releasing everyone from their box.
The fruit of Fate
The second major theme of Penguindrum is fate. Going back to the theme of abandonment, Kanba, Shoma and Himari were all fated to die. However, they "cheated"; Kanba was chosen by the fruit of Fate to survive, but he shared this fruit to Shoma, who in turn shared his fruit with Himari. They were "punished" for escaping the fate they were promised.
This punishment starts when Himari dies in the first episode. Having been revived by the Princess, Kanba pays another part of his fruit for Himari's life. But this was only temporary; once again Himari dies in the twelfth episode, which brings in Sanetoshi's intervention through his medicine. And yet again this wasn't enough, so Himari gives back her share of the fruit to Shoma.
This comes full circle in the final episode, when Shoma gives his share of the fruit back to Kanba (he no longer needs it). Ringo intervenes by transferring the fruit of Fate to Himari (so that Himari lives, at the cost of Ringo's life), but Shoma takes the punishment instead, resulting in the world after the transfer - Shoma and Kanba are Ret-Gone from existence, while Ringo and Himari survive.
In the end, the message this entails is that fate can be derailed, but it doesn't come cheaply. The net change is still 0; only one of the Takakuras were to survive. The only change was who was to survive. The sacrifice of the siblings in the end derailed the track of fate that Sanetoshi was planning. Sanetoshi did not bank on this selfless sacrifice that the siblings performed; this was the alternate route to Sanetoshi's Survival Strategy.
At the very end of it all, the root of all this is love. Selfless love for the family.
What is the Penguindrum?
The Penguindrum is still open to interpretation. In a sense, the Penguindrum could be the love and affection that the family needed. It could be interpreted, in a magical sense, the shared life force within the Takakura family. In a literal sense, it's the apple that Kanba was given and shared to his "siblings".
The "full circle" of fate makes up the "spinning" part of the title; everything was just the whim of the spinning wheel of fate that ties all the characters together. The "Penguindrum" from Kanba was shared with Shoma, who shared it with Himari, then in the end, the siblings' respective shares of the Penguindrum became whole again and finally given to Himari, the fated one of Shoma and Kanba's most beloved person.
An interpretation of the entire series
Courtesy of Eschaton on the TV Tropes Forums. This basically sums up all of the above points, while adding more about the Aum Shinrikyo connection to the series.