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 He shall bring them to the priest, who shall offer first the one for the sin offering. He shall wring its head from its neck but shall not sever it completely,