The first bird is killed by having its neck nipped, but not severed (see commentary on Leviticus 1:15). The fact that this comes before the burnt offering is significant. The sin offering must take place first in order to restore the worshipper to God. In other words, sin must be atoned for before there could be communion with God.
8 Il les apportera au sacrificateur, qui sacrifiera d'abord celui qui doit servir de victime expiatoire. Le sacrificateur lui ouvrira la tête avec l'ongle près de la nuque, sans la séparer;