So, we're starting with the first sacrament, the foundational sacrament: Baptism.
In the Catholic church, we tend to baptism babies. Of course some people are baptized as older children or even as adults, but baptism usually occurs when a baby is under a year old. It used to happen the day after birth, way back in the day--now it's usually a few months later.
We baptize babies for a few reasons: partially because Jesus said "Let the little children come to me", and also, to remove the stain of original sin.
Everyone is born with original sin--the sin of Adam and Eve. Baptism removes the sin, grants grace, and makes the person an "official" member of the Church. Once you're baptized, you can't be unbaptized. It's a permanent character that's embedded in your soul. Even if you never go into a church again, even if you decide you don't believe in God--once you're baptized, you're baptized forever.
Each sacrament has what's called matter and form. The matter is, to be blunt, the "stuff" involved. Here, it's water. You must have water. The form is what's said. "I baptize you in the name of the Father, the Son, and the Holy Spirit."
If you don't have both matter and form, then it's not a valid sacrament.
In an emergency, anyone can baptize, but usually, a deacon or a priest does it. An by emergency, they mean emergency. As in, premature baby born, in the NICU, etc.
Baptism begins a person's Christian life; thus, it's the first sacrament.