There are two kinds of blessings - invocative and constitutive. The second of these, a constitutive blessing, must be given by an ordained person (usually a priest or bishop), as the liturgical books specify. That attaches a blessing to a thing, which also brings a blessing to the one who uses it in a state of grace, and reserves the thing for sacred use, whether in a church, or for private use. Priests give “official” blessings - i.e. in the name of the Church - to persons, which can be repeated (like at the end of each mass). The priestly (or episcopal, given by a bishop, or apostolic blessing, given by the pope) blessing imparts graces to those who receive it in the state of grace (i.e. a baptized Catholic not in a state of mortal sin).
Priests cannot bless themselves in a priestly fashion - but they can bless themselves the same way a lay person can (making the sign of the cross). Priests have to go to other priests for confession, the Sacrament of the Sick, etc., also.
Hope that helps.