Let's start basic -
Buckling - Male goat under 1 year of age
**Doe - Female goat
Doling - Female goat under 1 year of age
***Wether - a castrated male goat
Kid - baby goat
A little more detailed -
Kidding - goat labor, a doe giving birth to her kids - she's kidding.
Freshening - When a doe kids and comes into milk, this is called a freshening. Freshenings are often counted for the doe's history sake, and to give a better estimated on the doe's milk yield. A first freshener (a doe freshening for the first time) will most likely not produce as much milk as a second or third freshener.
Lactation - The period through, or length of time, in which a doe is producing milk. The average dairy goat will have a lactation of 10 months, or 305 days.
Gestation - The length of time between breeding and kidding, through which the goat fetus' are growing and developing. Standard size dairy goats have a gestation period of 150 days. Mini's are closer to 148 days.
Udder - mammary gland - often incorrectly called milk bag, sack etc. Ideally a dairy goat's udder will consist of 2 halves, each with a single teat.
Colostrum - a does first milk - she will continue to produce colostrum for up to 24 hours after kidding. Colostrum contains immunities vital to kids survival not found in milk.
Caprine - the Latin name for goat - also as it's species classification / zoology term - Dogs are canine, cats feline - etc. Pronounced 'Cap-rine' -end sound is pronounced like rind as in mellon rind - but no 'd' sound at the end. Often incorrectly pronounced - 'Cap-reen'.
Disbudding - To remove horn buds from kids before they have pushed through the skin. Usually done with a hot iron on kids at 2-4 days of age.
De-horning - To remove horns after they have developed. Usually done surgically (Not a recommended procedure).
Rut - used to describe a bucks stress and behavior through the breeding season, or a period of time. A buck may start into rut in the fall and not come out of it until the spring, or whenever all the does are bred. A buck in rut will preform all of the normal rituals including urinating on his face, chest and legs, giving them their famous buck smell.
Drenching - To administer drugs, meds, other/etc. - *orally.
*Bucks are often also called Billies or Billy, and **does referred to as 'Nanny'. There is some debate as to which is the 'correct' term. However the difference in what you call them may be as simple as your location or mentor. Billy and Nanny seem to be more acceptable terms with meat goat herds. Where as many dairy show goat circles would take the terms as an insult describing a unhealthy or ill kept goat. More so it is just disregarded as slang.
***Wethers, not to be confused with 'Withers' the highest point on the shoulder. A goat's height would be measured from the withers down to the ground.