What kind of city and what kind of defenses does it have?
Give me those details and I can give you an answer.
Edit: However, let's assume early gunpowder levels of tech. You first start by surrounding the city, cutting off all food and resupply to them. Next you begin a bombardment. While not technically required a bombardment that lasts months would break the morale of the defenders and possibly destroy food. Theoretically after enough time has passed the city will surrender.
The problem here is that theoretically bit. What a lot of people (judging by fiction, and the answers below) don't seem to realize is that a siege without penicillin is almost as dangerous to the besieger. Sure disease can spread through the city, but it can just as easily get among your own soldiers. Without basic sanitation, clean water, and not too cramped living conditions (almost none of which are possible while conducting a siege) disease is likely to break out and begin killing off your own soldiers. In addition, it is a lot harder to supply an army than you might think. Within a week the army will have foraged everything in a large area around the city. From there you have to ship food in, a process that takes a long time, can be interrupted by enemy counterattacks (that city won't be their only one after all), and so your troops too can begin to starve. After a little more time the clean water runs out, unless you are next to a river and even then due to the way armies used the rivers before the modern day it is likely that water is undrinkable without boiling it first. So again, your soldiers die.
No, the correct answer here is: DO NOT BESIEGE A CITY!!!, It's a tactic of fools.
The city will likely outlast you simply because they will have more supplies built up and likely have reinforcements on the way. The way to defeat the city is to attack the countryside. If you can destroy the farms, poison the water, slaughter the population, etc. the garrison must sally forth to stop you. In open battle your chances of defeating the enemy are much higher. When the garrison is defeated you can take the city easily.
If you must begin a siege begin by taking control of all transportation and build decent supply lines. Ensure the latrines are dug far away from the soldiers and make sure all drinking water is purified. A steady supply of food and water will do more for achieving victory than any weapon will. And even in this case you would be better off battering a hole in their walls and offering a chance to surrender. Odds are the city will take it as they know to expect no mercy if they fight on.