How old and what weight does a heifer or cow have to be before she can get pregnant?

The optimum age is 15 months or more. She should also be weighing at least 60% of the average weight of the main herd or the body weight of a mature cow that is comprised of similar breeding to her. She must also have good pelvic area and good capacity in her hips and barrel area before she can have a calf. This is because some heifers can be still too small and too light to be bred at 15 months of age, and if they are bred when none of these criteria are met, there will be problems when she tries to give birth, resulting in dystocia. Most heifers that cannot meet these criteria in the first place are shipped to be used for meat production or, if they're lucky, bought by other producers who have a need for smaller heifers.

With a 285-day gestation period, she will calve out by the time she's ~24 months of age.

For cows, they are already mature females who are well past 15 months old, who have, supposedly, already had a calf or two. It's matter of whether they have calved out already and they have come back into normal cycling, or if they're not pregnant (already open), that determines whether they are ready to become pregnant again or not. However, a cow's ability to be able to breed back is determined by their weight in terms of body condition score: Cows can be a half a BCS score lighter than heifers (depending on what scale you are using), but they must be at a normal body condition of 5 to 6 (on a scale of 1 to 9) or 3 to 3.5 (on a scale of 1 to 5). They should not be too thin, nor should they be too fat prior to parturition. You should be able to see some of the bones on the cow, but they shouldn't be prominent or sharp like you would see with an emaciated cow. There should be a moderate fat covering over the ribs, spine, and pelvic bones. Rolls of fat on a cow is not good, as it could hinder the calving and lactating progress. Check out the following related questions on body condition score for cows and heifers.