Women have two ovaries, of course, and yet under normal circumstances only one ovum (egg) is released into the abdominal cavity each month. How is this coordinated?
A woman's ova must mature first before they are ready to be released. From birth to puberty, each ovum is surrounded by a single layer of granulosa cells and is termed a primordial follicle. This is more or less how a woman finds her ovaries just before puberty - filled with tens of thousands of primordial follicles.
Each month, some of these follicles are coaxed into further development by the action of two hormones produced in the pituitary gland: luteinizing hormone (LH) and follicle stimulating hormone (FSH). But here's the crucial point: only some of the primordial follicles - typically between six and twelve - develop further each month. These follicles are spread out across both ovaries.
Then, after about a week of stimulation (but before ovulation), one of the follicles begins to outgrow all the others. The others, sensing a winner, give up their chance at immortality and simply involute, leaving the only remaining one to eventually release its ovum. Which ovary (left or right) the winner comes from appears to be largely random - the ovaries aren't so well coordinated as to alternate sides or anything. For instance, the right ovum might produce the ova for three months in a row before the left gets a chance, or vice versa. It's like tossing a coin.
There are two related questions that are begging for an answer here. Firstly, if only one ovum is released each month, why do the other five to eleven of the primary follicles even bother to enter the race? A clue lies in the fact that as follicles develop, they begin to secrete oestrogen and progesterone. Apparently the body needs more than one cell per month in order to generate enough of these hormones - although it obviously needs only a single ovum for fertilisation purposes.
Secondly, what causes the other follicles to give up the race? As the follicles mature, they set up a positive feedback cycle whereby the oestrogen they produce sensitizes the cells to more FSH and LH, which further promotes their development and oestrogen secretion, which sensitizes ... (you get the picture). This process of exponential growth is therefore explosively quick, and so any follicle that edges ahead of the rest of the pack (even if through sheer randomness) quickly outstrips any rivals. It is postulated that this 'dominant follicle' soon secretes so much oestrogen that FSH and LH secretion by the pituitary is suppressed (high levels of oestrogen almost always do this). Starved of stimulation, the paucity of FSH and LH causes the other follicles to involute, whilst the largest follicle is still sufficiently sensitive to the lower levels of FSH and LH to continue its development.