Alligators are oviparous, which means that lay eggs.
More Info: There are only two species of alligators remaining on Earth, the American alligator (A. mississipiensis) and the Chinese alligator (A. sinensis). They both reproduce through sexual reproduction and they both lay eggs.
Courtship and Mating
Like many species, alligators perform courtship rituals prior to choosing a mate. Courtship generally begins in April if the temperature and weather conditions are right and in the peak of the season are initiated by the female. Males will attempt to attract the attention of a female with bellows and sub-audible vibrations. The pair will often be seen roughly pushing one another under water in an attempt to test the strength of the other. Male alligators are extremely aggressive during the mating season and will kill rival males. Female alligators are also extremely aggressive and will kill males that they do not find suitable.  The courtship generally lasts a month before the alligators choose mates. 
Though difficult to visually distinguish a male from a female alligator due to their hidden genitalia, they mate in much the same way that mammals do. Repeated copulation is common with the entire ritual lasting about 45 minutes from start to finish and generally occurs in early morning in deep waters. 
Where Do Alligators Lay Eggs?
Once mating has occurred, the female alligator will leave the water and look for a suitable spot to build a nest that is elevated for good drainage and protected from predators. Nest building and egg laying occur at night. Once she has found a suitable spot, she will begin to pile vegetation into a round mound and will usually attempt to create a nest multiple times before she finally settles on one. This nesting activity usually takes place throughout a two-week period.  The timeframe from copulation to egg laying varies but generally occurs in one to two months. 
In the manner that a woman is pregnant when she is carry a child, a female alligator is gravid when she is carrying eggs. The act of laying eggs is called oviposition
How Many Eggs Do Alligators Lay?
The female alligator will then lay her eggs on top of the pile and wait until they slowly sink down to the center. She will then cover the eggs, known as her clutch, with more vegetation. The clutch size generally averages between 35-40 eggs with a fertility rate of 70-95%. The number of eggs an alligator lays will vary with age and health of the female. Older, larger females generally lay more eggs. Due to predation, fewer than 70% will survive.  The mother will stay with the eggs and guard the nest. Following an incubation period of 65 days, the eggs are ready to hatch.
How Do Alligator Eggs Hatch?
The sex of the hatchling is determined by the surrounding temperature between days 20-35. The optimal temperature range is between 86°F and 91°F, which will produce a healthy number of both sexes. If the temperature falls below 86°F all females will be produced, temperatures above 91°F will result in all males. Temperatures much higher or much lower can cause abnormal development and high mortality.  When they are ready to hatch, the a few hatchlings will make a yerping sound triggering the others begin yerping. They begin pecking away at the surrounding shell with the egg tooth that has grown on the tip of their snouts for this very purpose. The mother will assist those that cannot extricate themselves by rolling the egg in her mouth. 
Test Your Knowledge
- If eggs are subjected to temperatures over 91°F between days 20-35, which sex will be produced?
- Due to predation, what percentage of eggs will produce live hatchlings?
- What is a group of eggs called?
- What is the act of laying eggs called?
- What does it mean when an alligator is gravid?
- Fewer than 70%
- She is carrying eggs
 University of Michigan-Animal Diversity Web
SRAC Fact Sheet
Alligator Production, Breeding, and Egg Incubation
eNature; Burton, K
Alligator Egg Fertilization
 University of Florida Extension
Living with Alligators: A Florida Reality
Savannah River Ecology Laboratory
American Alligator (Alligator Mississippiensis)