First you need a female. Open her page and if she is ready to be breed a Breed Button will be in the upper Right corner of her page. If she is not, it will say 'pregnant' or a number. If it has a number, that number must go to zero before she may breed again. If the button is there click the 'breed' button.
A list of studs will come up. Your studs (assuming you have some males too) will be on the page, it shows the first 50 of your males, the search tools at the bottom can be used to search your own or public studs.
If you find one you might want to use, open his page (by clicking on his name) in another window, (Rt. Click on his name and select Open Link In New Window) Check out his stats and offspring if you want to see what he has produced. When you are done just close this window.
If you see animals listed in Red they are closely related to your female and should not be bred to that girl, you can breed to them but you will take a big quality loss for inbreeding.
Click the Breed link for the one you want to breed to your female. The next page will tell you if the breeding was successful or not. If you used a public stud you will be charged the stud fee that was listed, your own males of course are free for your use, and BTW putting your males out to stud can generate income for you. Always need money even in a game!
If it takes you will have to wait 3 days for the birth in most breeds, the egg layers are 3 hours to lay the egg and then 3 days for it to hatch. Rabbits and Mice I think are 3 hours till birth and 3 days of being "babies".
When she is ready to give birth you will get a notice/link on your Main Ranch Page announcing that "whoever" is ready to give birth. Click on this link on your main ranch page, or Rt. click and open in a new window, it will give you the birthing list, after reading it you can close it.
The newborns will now be waiting for you in your Open Range Pasture, before anything else click Animal Hospital. This is the Vet link, send all that show up to the vet, this by the way should be done at least once daily to keep your herd in top condition, anyone needing vet services will show up. If none show up then all is well!
After the vet you can open the newborn's pages and DNA and Name them.
The Mothers always need vetting after giving birth!
Breeding can also be done from the Male's page, but it seems easier (at least to me) to breed from the Female's page.
Currently the newborn don't seem to need vetting after being born, this might be a bug, I'm not sure so always check in at the vet before doing DNA's as they might need it at some point again!