Thank you for raising the important issue. There are some important distinctions between round and square matzot. To explain I need to provide a little background information.

The Torah states, “And you shall guard the matzot” (Exodus 12:17). The meaning appears to be that one must guard his grain and flour that they not come in contact with moisture and turn into leaven before he bakes them into matzah. This, however, seems superfluous. All the foods we eat on Passover must be checked that they are not and do not contain chametz. Why did the Torah single out matzah?

Based on this, the commentators explain that a special degree of “guarding” is required for matzah. Whereas for all other foods it is sufficient to ascertain that they contain no chametz, matzah must be specially guarded for the sake of the holiday. This is a positive obligation – to guard the grain and flour that they not become chametz, and to later bake them into matzah specifically in order to have matzah for the Passover obligation. We call such matzah “shmurah matzah” – which literally means “guarded matzah.”

Shmurah matzah is required only for the Seders – specifically for the matzah we eat at the beginning of the meal and for the afikomen we eat at the end. For the rest of the Seder meal and for the rest of Passover, “ordinary” matzot are sufficient.

To be considered shmurah matzah, the wheat should preferably be guarded from the time it is harvested. According to some authorities, however, it is sufficient if it is guarded from the time the grain is ground into flour.

This finally brings us to round versus square matzot. Round matzot, which are handmade, are generally made from wheat which had been guarded since the time it was harvested. This is strongly preferred for the Seder. (Make sure to buy from a reputable seller, with kosher certification printed on the box.) Square matzot, made by machine, are typically made from flour guarded only from the time it was ground. These are generally sold as “18 minute matzot” – meaning that although they were not specially-guarded from the time of harvesting, they were baked within the proper time limit and are guaranteed not to be chametz.

In truth, as we explained, there are authorities which consider guarding the flour from the time it is ground sufficient. Thus, those who take ordinary “18 minute matzot” for their Seder do have basis for their practice.

It is also possible to purchase “machine shmurah matzah.” This is matzah made from wheat guarded from the time it was harvested, but which was baked by machinery rather than by hand. According to many, this would be usable for the Seder as well. However, some authorities hold that the notion of making the matzah for the sake of the mitzvah can only be fulfilled if the matzah is produced by hand, since machines cannot have any intent. (For machine matzot, the machinery is operated by a person who turns it on for the sake of the mitzvah.)

Although shmurah matzah is only required for the Seder, there are some authorities who state that every time we eat shmurah matzah during Passover it is an extra fulfillment of the mitzvah. Thus, some meticulous individuals eat only shmurah matzah (hand-baked or machine-made) the entire holiday.

(Sources: Shulchan Aruch O.C. 453:4, Mishna Berurah 21, 475:45, R. Eider Halachos of Pesach pp. 212-215.)