Note what Fr. Tommy Lane has to say about it:
On the façade of the basilica there is an inscription in Latin which reads, “the mother and mistress of all churches of Rome and the world.” One might think St. Peter’s Basilica is the head of all the churches but in fact it is the Basilica of St. John Lateran. Every bishop has a cathedral and the Pope’s cathedral is the Basilica of St. John Lateran not the Basilica of St. Peter.
Yes, St. John Lateran is the Pope's church, and the "mother of all churches". (That sounds a little naughty, doesn't it?)
|Triclinium of Leo III|
The wooden altar on which St. Peter celebrated Mass while in Rome is inside the main altar. The heads of Sts. Peter and Paul were once believed to be inside busts above the main altar. Part of the table on which the Last Supper was celebrated is said to be behind a bronze depiction of the Last Supper. At one time the Holy Stairs which is nearby was also in the Lateran, the stairs in Pilate’s house on which Jesus is said to have walked during his trial. It is a marble stairs and is now covered with wood to protect it. Pilgrims ascend the stairs on their knees contemplating Jesus’ Passion and on the way up drops of blood may be seen on the marble stairs beneath protective glass. The stairs was brought to Rome by Constantine’s mother St. Helena.
|Scala Santa, or Holy Steps|
This is why the universal Church today celebrates this place: its historical significance, its importance to the papacy, its beauty, and the fact that it is (even if we've never been there) our church.