From Women for Faith & Family:
HOLY THURSDAY is the most complex and profound of all religious observances, saving only the Easter Vigil. It celebrates both the institution by Christ Himself of the Eucharist and of the institution of the sacerdotal priesthood (as distinct from the "priesthood of all believers") for in this, His last supper with the disciples, a celebration of Passover, He is the self-offered Passover Victim, and every ordained priest to this day presents this same sacrifice, by Christ's authority and command, in exactly the same way. The Last Supper was also Christ's farewell to His assembled disciples, some of whom would betray, desert or deny Him before the sun rose again.
...The evening Holy Thursday Liturgy, marks the end of Lent and the beginning of the sacred "Triduum" ("three days") of Holy Week, which culminates in the Easter Vigil, and concludes at Vespers on the evening of Easter day. The Mass begins in the evening, because Passover began at sundown; it commemorates Our Lord's institution of the Holy Eucharist at the Last Supper. It also shows both the worth God ascribes to the humility of service, and the need for cleansing with water (a symbol of baptism) in the Mandatum, washing, commemorating Jesus' washing the feet of His apostles, as well as in the priest's stripping and washing of the altar. Cleansing, in fact, gave this day of Holy Week the name Maundy Thursday.
The action of the Church on this night also witnesses to the Church's esteem for Christ's Body present in the consecrated Host in the Adoration of the Blessed Sacrament, carried in solemn procession to the flower-bedecked Altar of Repose, where it will remain "entombed" until the communion service on Good Friday. No Mass will be celebrated again in the Church until the Easter Vigil proclaims the Resurrection.
And finally, there is the continued Adoration of the Blessed Sacrament by the people during the night, just as the disciples stayed with the Lord during His agony on the Mount of Olives before the betrayal by Judas.
Before the feast of Passover, Jesus knew that his hour had come
to pass from this world to the Father.
He loved his own in the world and he loved them to the end.
The devil had already induced Judas, son of Simon the Iscariot, to hand him over.
So, during supper,
fully aware that the Father had put everything into his power
and that he had come from God and was returning to God,
he rose from supper and took off his outer garments.
He took a towel and tied it around his waist.
Then he poured water into a basin
and began to wash the disciples’ feet
and dry them with the towel around his waist.
He came to Simon Peter, who said to him,
“Master, are you going to wash my feet?”
Jesus answered and said to him,
“What I am doing, you do not understand now,
but you will understand later.”
Peter said to him, “You will never wash my feet.”
Jesus answered him,
“Unless I wash you, you will have no inheritance with me.”
Simon Peter said to him,
“Master, then not only my feet, but my hands and head as well.”
Jesus said to him,
“Whoever has bathed has no need except to have his feet washed,
for he is clean all over;
so you are clean, but not all.”
For he knew who would betray him;
for this reason, he said, “Not all of you are clean.”
So when he had washed their feet
and put his garments back on and reclined at table again,
he said to them, “Do you realize what I have done for you?
You call me ‘teacher’ and ‘master,’ and rightly so, for indeed I am.
If I, therefore, the master and teacher, have washed your feet,
you ought to wash one another’s feet.
I have given you a model to follow,
so that as I have done for you, you should also do.”