There are a lot of misunderstandings about the Catholic use of Holy Water. People sometimes think we believe it has magical properties or something along those lines. But holy water is merely water that has been blessed to use for a divine purpose.
When we think of water, we think of purity, cleansing, refreshment, quenching. It symbolizes all these things and more to us when we use it for baptisms and for blessings.
This article explains that "the ritual use of this precious substance is ancient and rooted in the Old Testament. When the Israelites entered the Temple, they had to undergo purification by immersion in a mikvah...These ritual purifications by water prefigured Christian Baptism, which we recall when we bless ourselves (cross ourselves) using holy water upon entering our churches."
There are so many bible stories that include water as a principal element, beginning in Genesis and the Creation account, to Noah and the flood, to Moses and the water springing forth from the rock (which happens to be what the painting above is depicting). All of these portray an aspect of water as essential and life giving, thus prefiguring baptism.
Returning again to the article linked above, we learn that "holy water is also used by the priest to sprinkle the people before the beginning of Mass." [This can also happen during mass throughout the season of Easter, also recalling our baptism.]
"He will have a pail-like vessel called an "aspersory" to hold the holy water, and into this he will dip a stick called an "aspergillum" (or an "aspergill"). The aspergillum has holes in it to catch droplets of the water, and the priest's swinging it toward the congregation causes the drops of water to fly out. This blessing of the congregation before Mass is called "Aspérges" and the accompanying, chanted words come from Psalm 50:
Aspérges me, Domine, hyssopo, et mundabor: lavabis me, et super nivem dealbador. Miserére mei, Deus, secundum magnam misericordiam tuam
Thou shalt sprinkle me, O Lord, with hyssop and I shall be cleansed; Thou shalt wash me, and I shall become whiter than show. Have mercy on me, O God, according to Thy great mercy
Here is just a small sample of other verses from scripture related to water:
In an earthen vessel he shall meanwhile put some holy water, as well as some dust that he has taken from the floor of the Dwelling. (Num 5:17)
This is what you shall do to them to purify them. Sprinkle them with the water of remission; then have them shave their whole bodies and wash their clothes, and so purify themselves. (Num 8:7)
But when they came to Jesus and saw that he was already dead, they did not break his legs, but one soldier thrust his lance into his side, and immediately blood and water flowed out. (Jn 19:33-4)
Jesus answered and said to her, "Everyone who drinks this water will be thirsty again; but whoever drinks the water I shall give will never thirst; the water I shall give will become in him a spring of water welling up to eternal life." (Jn 4:13-4)
And finally, here's another great video from That Catholic Show discussing the importance of water to Catholics.