There are so many things worth seeing in Berlin, I am just going to focus on the things I saw myself.
The Berlin Wall (Die Berliner Mauer)
The Wall together with the most tourist landmark - Checkpoint Charlie and the museums on this topic are definitely things worth visiting. If you are into such things you can even acquire a visa allowing free passage through the checkpoint to the other side and also buy a peace of the famous wall to take home with you. Both are available at rather high prices.
Even after seeing it in person one can only begin to imagine how it looked back then when the Wall was still performing its task. A few sections are still standing today. For a more genuine impression I recommend you check out one of the non centrally located sections of it.
Museums and Galleries
There are countless museums opened around the city. One can easily find at least a couple to his liking. Of those visited I just have to point out the DDR Museum and the Deutches Technikmuseum.
Both of these two house a vast number of expositions and if you want to pay any attention to detail, you should reserve at least half a day for each one of them. Even then I suggest you make a list of what to see first. If you don't run out of time, your feet might let you down in the process.
Both museums offer a highly interactive experience to their guests. I must say they do a great job!
Nightlife, Club Scene and Festivals
Berlin offers a great variety of socializing opportunities. I had a feeling there is always a club opened nearby. Many of them are building their reputation on some kind of a weird or even bizarre offer. In some cases their reputation is based on their setting inside some old power plant or something like that, in other cases it is all about the happening.
It is always worth asking around for something of your liking. Bizarre does not necessarily go together with enjoyable!
If you find the right club and enough energy, it might just happen you start the party late in the afternoon and don't stop until one next day... and by one I mean one o'clock in the afternoon!
In the warmer months many festivals also take place in the opened. Love Parade is probably the most famous of the bunch.
City ZOO and Aquarium
Zoologischer Garten Berlin houses the most diverse range of species in the world. Among its inhabitants is also the captive-born celebrity polar bear Knut.
Although the ZOO itself is still just a big cage it is well worth visiting if this is your thing. However if you have a choice, you should perhaps not plan a visit in winter time. During other seasons you will be able to see more animals out in the open.
A large city aquarium also features the AquaDom - the World’s Largest Cylindrical Aquarium. At 25 meters tall, and 11 meters wide, it is the largest acrylic glass cylinder in the world and contains 900,000 litres of sea water. Divers jump in every day to clean it and to feed the 56 different species of fish living inside.
However cold it might get during winter in Berlin, it always feels good to warm up with a cup of mulled wine (yes, that's Glühwein if you were wondering).
Usually there is also a sausage stand nearby offering many variations of this traditional German food.
I guess I should also mention some other things... but this time I will leave it to you. So please point out some of the highlights you liked the most.