In Catholicism, what does it mean for a priest to bless an object or person?

Blessing of an item is sort of a "package for prayer". The blessing is essentially a prayer for something to remind us Christ and our faith. For example, water reminds especially baptism and everything Jesus did about it in the events recorded in Bible (not only in New Testaments, there are some images pointing to baptism in Old Testament, like passage through Red Sea), in history and in our lives. Another example: a blessing of a cross means a prayer for its wearer to be connected with Jesus, who died on a cross, with all its consequences.
EDIT: Similarly, blessing of a person is a prayer for someone aiming not just to a "one-shot" benefit, but to a longer benediction - so that the person gets closer and closer to God, performs well in their job or just enjoy good luck in general.

Catholic Church take all the blessings of water, crosses and other things and people by priests as sacramentals - they are not powered by God's promise as directly as sacraments (Eucharist, baptism etc.), but prayers of whole church somehow "power" the sacrament. Prayer is "better" when two or three agree on it - but in sacramentals, all the Body of Christ agree on it, even though most of us Christians (or at least Catholics) are not present or even don't know about it.

And what exactly changes? It's the same as with any other prayer - a clear miracle may occur, but usually nothing apparently changes. But we know God heard the prayer, and we believe that He will answer somehow.

Blessing expresses the basic movement of Christian prayer: it is an encounter between God and man. In blessing, God’s gift and man’s acceptance of it are united in dialogue with each other. The prayer of blessing is man’s response to God's gifts: because God blesses, the human heart can in return bless the One who is the source of every blessing.

(Catechism of the Catholic Church, paragraph 2626)

Blessing is a divine and life-giving action, the source of which is the Father; his blessing is both word and gift. When applied to man, the word "blessing" means adoration and surrender to his Creator in thanksgiving.

(Catechism, paragraph 1078)

In other words, the Catholic Church uses the word bless in two ways. We say that God blesses us when God gives us a gift (material or spiritual); in return, we bless God when we adore him as the source of all gifts.

When we ask a priest to bless an object, what we are actually doing is asking him to ask God to bless us in our interactions with the object; that is, we're asking God to make the object not simply an object, but a spiritual gift to us, an occasion of grace for us.

For example, a simple blessing of an Advent wreath is found on the website of the US Conference of Catholic Bishops. The prayer of blessing includes the text

Lord God,
let your blessing come upon us
as we light the candles of this wreath.
May the wreath and its light
be a sign of Christ’s promise to bring us salvation.

In other words, asking God to bless the wreath is actually asking God to bless us as we interact with the wreath, to give us the grace of remembering God's promises that He will save us.

That's how blessings work in general. To ask the priest to bless an object is to ask him to pray to God, requesting that God give us the gift of interacting with Him (giving us His grace) as we use this object.

Blessings of a person work similarly. We usually don't bless a person in an absolute sense, but in regards to a role they have—for example, one wouldn't bless Joe Blow as Joe Blow, but if Joe were a student, we might bless him as a student.

Again, there is a good example on the USCCB website: in this case, a blessing of parents expecting a child. There is a reading from Scripture (Samuel 1:19–20,1:24–2:1, recounting Hannah's expectant hope for the birth of her son), and then a prayer including the following text:

Hear the prayers of N. and N.,
Who await the birth of their child.
Calm their fears when they are anxious.
Watch over and support these parents
and bring their child into this world
safely and in good health,
so that as members of your family
they may praise you and glorify you
through your Son, our Lord Jesus Christ,
now and for ever.

Again, it's asking for God's gifts of calmness, patience, and hope for the parents, and of good health for parents and child, so that in return all of them may know the Lord's goodness and praise and thank Him for that.

It is not necessary for a priest to bless an object or a person; anyone can bless an object. And since anything can be a proximate occasion of God's grace (that is, God can use any thing or any occasion to pour out grace on us), it shouldn't be surprising that the Catholic Church has historically created prayers of blessing, and even liturgies of blessing, for nearly anything or anyone imaginable. I've asked God to bless my phone and my computer keyboard!