Welcome to Maseches Kilayim. The prohibition of kilayim - the mixing of different kinds of things with one another - covers several distinct areas of halachah:
Kilei zeraim - the mixing of different seeds with one another, which is the subject of the first three chapters of the tractate. This prohibition applies in Eretz Yisrael alone.
Kilei hakerem - from chapter 4 through chapter 7: the mixing of other trees and vegetables together with a vineyard. By Torah law this is forbidden in Eretz Yisrael alone, and by rabbinic law outside Eretz Yisrael as well. Included in this category is also the prohibition of grafting - one may not graft one tree onto another, and this prohibition applies by Torah law even outside Eretz Yisrael.
Kilei behemah - in chapter 8: the pairing of two animals with one another, both regarding working them together and regarding mating them.
Shaatnez - in chapter 9: the mixing of wool and linen together, which is forbidden by Torah law both in Eretz Yisrael and outside it.
Chapter 1, Mishnah 1:
The Mishnah opens by listing groups of seeds that one might have thought are different species, and would therefore be kilayim with one another and forbidden to mix; in fact, however, since they are considered a single species, they are not kilayim with one another. The list is presented in pairs: the Mishnah lists one pair and then another, each of which is not kilayim within itself.
"HaChittim vehaZunin" - wheat, and zunin, which is an inferior type of wheat. These two types of wheat are not kilayim with one another, and it is permitted to mix them together.
"HaSe'orim veShibbolet Shual" - barley, and shibbolet shual, which is a type of oat. Although they appear to be different species, they are considered a single species with regard to kilayim, and therefore it is permitted to mix them.
"HaKusmin vehaShifon" - these too are considered a single species, and may be mixed together.
"HaPol vehaSapir" - the broad bean, and another type of bean that closely resembles it. These too are considered a single species.
"HaPurkedan vehaTofach" - two types of legumes for which we have no precise translation of their names (and even if we did, it would not make much difference for our purposes), and they are not considered kilayim with one another.
"UPol haLavan vehaSha'it" - the white bean, and the sha'it, which is another type of legume that some translate as the cowpea. These too are not kilayim with one another.
In summary: as the Mishnah concludes - "einam kilayim zeh bazeh" - each of these pairs is not considered kilayim with its partner. Mixing one of them with a member of a different pair, however, is kilayim. It emerges that only within each individual pair does the law of kilayim not apply.