IMO, 48 hours is far too short in London. I think you need weeks, months, years to actually know London to its very core. But if you're only allowed 48 hours in this wonderful city, there's some mandatory places/things that you must see/do. I'll throw in some of the typical touristy things, as well as some of the back road activities that you maybe normally wouldn't get to.
Things to Do
Big Ben / Westminster Abbey / Houses of Parliament / Downing Street / Horseguards Parade / Trafalgar Square
These attractions are all really close to each other, which is why I included them in one. They're all part of a very touristy area of London called Whitehall in the Borough of Westminster (London is broken up into boroughs, sort of like counties). You can go up in Big Ben, tour Westminster Abbey (it's a little costly), visit Parliament, wander around outside Downing Street (where the PM lives), which is right next to Horseguards Parade, and find yourself in the heart of Trafalgar Square. All of these activities can be costly, so watch your wallet. Going to church at Westminster Abbey is free, so if you don't necessarily want a guided tour but want to see the church, maybe consider going for a mass.
This vibrant neighborhood in central London has lots to do and see, and is particularly known for its nightlife. It's the "gay neighborhood" of London, so wander this way during Gay Pride weekend and you'll be sure to disappear into the crowds forever. Check out Carnaby Street which is a famous shopping street on the West End.
Another vibing neighborhood, CG is home to tons of shops, restaurants, and street performers. Also A LOT of tourists. But it has a cute little market that's always buzzing.
London Eye / South Bank
The London Eye is located south of the river Thames on what is called the South Bank. Often during the summer / winter months, there is some sort of festival going on near the Eye. From Christmas Markets to comedy festivals, there's always something going on. There's lots of good places to eat around the South Bank as well.
Brick Lane / Shoreditch
Brick Lane is well known for its excellent Indian food and street art. It empties out into Shoreditch, an edgy, fun neighborhood with tons of street art, bars, clubs, and more underground shopping. There's also lots of little markets around there. Check out the Brick Lane Market on Saturdays. Be sure to keep an eye out for some Banksy paintings!
A Londoner's worst nightmare, Leicester Square (pronounced Lester) is the tourist hub of London. It's sort of the Times Square of London, with all the West End productions being performed here. The prices are unbelievably high, but you can't go to London without having gone to Leicester Square... It would just be a crime.
St. Paul's Area
The area around St. Paul's Cathedral is full of rich bankers working 70 hours a week and bringing home massive pay checks, and is cool to go to particularly during after work hours. When the weather is nice, all the bankers in their suits go to the pub at around 6 pm and the after work vibes are buzzin'. The Cathedral itself is cool. If you don't want to pay for it, you can go for the Eucharist and walk around after... Not that I'm endorsing that... But yeah.....
An obvious one. They open the state rooms for tours during the summer. I'd avoid visiting during changing of the guard; it's packed with people and difficult to see anything. Try going in the early evening when the tourists will die down. Wander around St. James's Park when you're done!
DO THIS. Take a few hours and walk up and down the Thames on the north and south banks of the river. This is a great way to get a feel for the city. It's got some beautiful views and you'll get to walk a good portion of London.
There's not much to do around the area, but Tower Bridge is another London must see. I recommend skipping the Tower of London if you're on a budget / time crunch. It's decently cool but pretty costly and takes up a large chunk of your day.
London is well known for its market, and contrary to popular belief, the food here isn't so bad. London boasts foods from all over the world, so you'll rarely be hungry if you're trying all the different foods it has to offer. You definitely won't have time / room in your stomach for all of these markets in just 48 hours, but definitely make a stop by 1 or 2 of them.
My favorite market. It could take a whole day to get through if you felt like it. They've got food stalls from all over the world, interesting souvenirs and art pieces, clothing, fried Mars bars, whatever you want. Be sure to check out the store CyberDog for an interesting experience.
London's most famous food market, Borough Market, has lots of options for eating near London Bridge. The market is open fully from Wednesday - Sunday.
Spitalfields is not much of a food market (although it does have some food), but it has lots of craft activities, small vendors, and cool vintage clothing for sale. It's not far from Shoreditch so you can pick up a drink on your way home.
Columbia Road Flower Market
London's most beautiful market (in my opinion) is flowing with flowers and colors every Sunday. In addition to flowers, you'll also find pottery shops, and small independent galleries. Get there early because things start to wind down at 2pm!
Brick Lane Market
Brick Lane Market has kind of an edgier feel. The market itself is comprised of a lot of little markets, like the Backyard Market, Sunday Up Market, and Vintage Market, but they're all relatively close together so you can walk quickly between each one.
Urban Food Fest (Shoreditch)
Every Saturday, a parking lot in Shoreditch is turned into an Urban food market. It's open till Midnight so it's perfect for drunk munchies if you're drinking in the area.
Where to Drink
London's nightlife is special, with lots of cool neighborhoods to wander through. I recommend you don't stay in one place, hop around so you can maximize your experience! Pubs typically close between midnight and 1am, so bear that in mind on your night out.
My stomping grounds, there's lots to do around Camden at all hours of the day/night. Camden attracts a small population of the "gothic" culture, so you can find a few goth bars like Underworld or The World's End. If that's not your scene, try out Proud for a clubbing atmosphere, the Edinboro Castle for a pub/beer gardeny feel, or the Blues Kitchen for live music / dancing / American food.
In Soho you'll find lots of gay bars and clubs, in addition to your everyday club. This area of the city is always vibing, as it includes Chinatown as well. My personal favorite is O'Neill's (the best dancing, the best drink deals, fun crowd) which is a guilty pleasure. For gay bars, head to Heaven and G-A-Y.
As I mentioned quite a bit above, Shoreditch is probably a combo between Camden and Dalston in terms of crowd, drawing a large hipster crowd. But there's so much to do in Shoreditch that it makes for a great night out. The Queen of Hoxton has a rooftop bar perfect for evening summer drinks. Cargo has more of a club vibe, but also has a large outdoor garden (complete with a Banksy painting). I highly recommend theLondon Gone Wild bar crawl here.
Attracting a large post-grad crowd, Clapham (in South London) is known for it's large bro (lad) population. It's still a great place for an evening out, with tons of bar options for both younger and older crowds. Check out Adventure Bar (which has great happy hour deals) for interesting cocktails, or hit up Infernos if you feel like getting hammered.
Probably the edgiest neighborhood in London, be prepared to be swarmed by smelly hipsters and craft beer and dirty jazz bars. But that's what makes this place so cool. Check out the Servant Jazz Quarters, they have delicious cocktails and live music. The Dalston Jazz Bar is also a blast for a night out. It's small and intimate, the drinks are decently priced, and the music is on point.