A quick and dirty run down:
Adult females will be much longer and broader in the pelvic fin region; I like to think of it as birthing hips and It is easier to see when you are looking at them from above. If you cant sex them, it will become clear once they reach the breeding age, I have read that it takes 8 or 9 months, they will begin a fattening up period which last about a month. The females will look obese and full of eggs even after 24 hours of not feeding.
It depends on the species but the easy to breed corys (~150 described cory species with pepper and bronze being reported as the easiest to breed) will be successful with a pH 6.5 to 7.5 and in either soft or hard water. Temperatures range is wide as well but 75 to 77 degrees is recommended.
I keep my corys with Neon Tetras and have had no problems other than the 5 or so eggs they may eat. The Tetras do not seem that interested in the eggs or the food sitting on the bottom to cause a problem.
Once the corys reach the appropriate age, you will need to make sure you are feeding them well. I use an assortment of sinking wafers, micro pellets and frozen foods (blood worms and brine shrimp). They are slow eaters and you will need to make sure they get enough food without polluting the water. Keep in mind that aggressive feeders like guppies will likely cause them to not get enough food to prepare for breeding (fattening period).
Once the female/females are fattened, which can take about 3 or 4 weeks, you can promote them to begin spawning by doing a significant water change and adding cooler water. I do a weekly 50% water change and when trying to promote breeding, I add water that is about 5 degrees. Usually, some time the next morning (12 or 15 hours later) the fish will begin to breed. Eggs are laid on the glass, plants and filter.
Egg care and hatching:
Ideal care would be to move eggs ( carefully with a razor) to a small container like a tip and pour (the plastic cup some pet stores use to hold fish while netting). Hang it over the side of the tank but with it inside of the water. This will keep the water inside the proper temperature. Use water from the tank where the breeding occurred as it is said to contain hormones that improve egg hatching. Clean the Tip and Pour daily and refill with breeding tank water. Keep an airstone inside of the tip and pour until the eggs hatch which will take 5 or 6 days. It is highly advisable to use an egg fungicide like Maroxy but make sure that it is completely removed form the tip and pour prior to hatching and not to contaminate the tank that the tip and pour is inside. Once they have hatched they will not need to be feed for 2 to 4 days as they will feed on their egg sac and can be moved to a cycled tank of there own. Once they become free swimming, no longer weighed down by an egg sac, you will need to feed them either finely crushed micro pellets or Hikari first bites or similar small foods.
Im sure planetcatfish.com has some well writing articles if you are looking for more information. I would suggest at least 5 of the same species and the biologist in me would also like to recommend not hybridizing corys.