The Marienplatz is the heart and centre of the city -- tourists wander or collect to see the glockenspiel on the Neues Rathaus, locals eat their lunch or read the paper. The square started out as a corn market, later became an execution site and jousting arena, and then, in 1315, Emperor Ludwig the Bavarian decreed that the square should never be built on. That seems to have been honoured (save for a statue in the centre outside the Rathaus -- see picture below) and it's a pedestrianised, fairly leisurely spot with some very interesting buildings and sights.
The Neues Rathaus (New Town Hall) fills the whole of the northern edge of the platz. Though it looks for all the world like a true Gothic, it dates from only 1867-1909. Its tower is 80m tall and can be accessed via a lift just inside and to the left as you pass from the platz into a mini-beer garden. Tickets are sold on the 4th floor (€2pp) and then you head up to the 9th floor for a pretty good view of the centre of town. The famed glockenspiel is however a big disappointment -- it's way, way up (if you're into that sort of thing, bring binoculars!) and very little happens when, at 11am, noon, 5pm and 9pm, the chimes ring out and two-tiered figures "burst" into life. Most striking on the town hall is the intricacy of the many, many figures which line the walls and roof (see photo below). Also look out above the arched stone entrances for the plaques featuring the monk who remains the symbol of Munchen.
To the right of the new pretender is the Altes Rathaus, a far more elegant (though confusingly younger-looking) whitewashed building dating from 1475. Its beautiful square tower apparently houses a toy museum ("Spiegelzeugmuseum") but it's otherwise inaccessible. Musicians often busk underneath its white right-hand arch. In the centre of the platz outside the Neues Rathaus is a statue of the Madonna with, on each of the four corners of the plinth, mini-cherubic warriors doing battle with representations of heresy, famine, war and plague.
To the far right, just beyond the square, though its spire is visible, is St Peter's, the oldest church in the city (records date from 1169), beloved by locals. Also, on the way into the platz from the south, you'll pass a wooden door into the Burgersaal, a former meeting place-cum-church. Downstairs is a low-ceilinged chapel, dedicated to Father Rupert Mayer (whose lies entombed there) who was taken away by the Nazis as a vocal troublemaker but reinstated after vociferous complaints by his flock (sadly he died in 1945); upstairs is a magnificent, high-ceilinged auditorium with frescoed ceiling and beautiful organ and altarpiece.
Last but by no means least, on Frauenplatz is the Frauenkirche, fast becoming the symbol of the city -- as well as going up the south of the two oxidised copper onion-domed towers (€3pp, every day save short closure between 3 and 3:15pm on Saturdays for bell ringing), by climbing a spiral staircase and then ascending in the lift, do go inside. It's a fairly minimalist interior, badly damaged in WWII (there are some photos to show how the walls were blown in and you can see from the stained glass windows the level of necessary reconstruction), though it trades in part off the "Devil's footprint." There are various stories for tourists -- my favourite one goes that the church's architect contracted to sell his soul if he failed to build a church in which no windows could be seen -- on delivery day, Satan came to collect, only to be frustrated when the architect demonstrated that, from a certain angle, no window could be seen -- in fury, the devil combusted, leaving a black footprint in the marble floor... -- these days, there are of course many and varied attractive windows.
More striking than any of this is the width of the church and the straight, gleaming white columns -- most of Munich's churches have a pleasing plainness to them (see in particular the Theaterkirche and Ludwig's kirche, both on Ludwidstrasse); there is magnificence and intricacy in design but it's not overwhelmingly gilded and ornate (though sights like Nymphenburg Palace more than make up for that!)
There are lots of sandwich shops round the platz, selling great bread/rolls or open sandwiches for an in situ lunch or which make it a good place to stock up before heading out for the rest of the day, if you're in the market for a picnic and haven't tracked down the Victualenmarkt. For dessert, there are unexpectedly good-value sorbets or ice creams to enjoy in warm weather. There's also a tourist information office.