What are the Psalms?
The Psalms are the prayer book of the Old Testament which were written during the time of King David. In Judaism, even today, the psalms are sung as part of their worship. Jesus Himself prayed the psalms and therefore we use them in our liturgy even today. Most notably, the psalms are seen in the Introit, the Gradual, the Offertory verse and the Communion verse of the mass. 
What is the Gradual?
The Gradual is the chant between the readings of the mass. It comes from the Latin gradus which means “step” because these chants were once said on the step of the ambo or altar. It was in earlier ages in the form of a responsorial psalm with a cantor and choir singing. Later, it developed into a couple verses usually from the psalms with the last verse being a repetition of the first. Today, most Graduals are even shorter. The Alleluia and the Sequence are later developments from this gradual. When the gradual is chanted, it is usually very lengthy (through beautiful) and the sacred ministers sit down. In a certain sense, the Gradual is our response to what is revealed by God. We worship Him and share our thoughts and feelings with the prayers of the Church. It is time to meditate on what was just read and prepare oneself for the Gospel. 
What is the Alleluia?
Alleluia is a Hebrew word which means, “Praise the Lord.” After the Gradual, two alleluias are said followed by a verse which more fully expresses or develops that Gradual followed by one more alleluia. This text is usually a verse from another part of scripture. The Alleluia was introduced into the Roman liturgy by St. Jerome from the Byzantine liturgies of Jerusalem. 
Is the Alleluia always said?
No, during funerals and days of penance, the Alleluia is usually silent. During Lent and Advent, it is even replaced by a tract. A tract is a prayer of solemn and penitential character usually taken from the psalms. Tracts tend to be long and only sung by one singer to show the sorrow in the chant. They express distress and contrition as well as confidence in the mercy of God. On some days the celebrant is required to genuflect during the tract to show his penitence. 
What is a Sequence?
The Sequence is an extension of this chant between the readings. This chant is the fullest and most intense expression of the sentiments found in the previous chants. The Sequences were first written in the 10th century as a means to remember the chant of the Alleluias. (Since writing music was a challenge at the time and the final notes of the Alleluias would extend through a single held syllable, placing words to the chant was a mnemonic device.) These Sequences became so popular that they entered into the liturgy every Sunday of the year. However, by the Council of Trent, Pope St. Pius V felt that it unduly lengthened the time between the readings and not all sequences were of good quality. So, he eliminated many of the sequences that were once used by Church and preserved the best ones. The sequences still used today are the Victimae Paschali  (which is recited on Easter), Veni Sancte Spiritus (from Pentecost), Lauda Sion (from Corpus Christi), Stabat Mater (on the Seven Dolors of the Blessed Virgin) and the most well-known Dies Irae (on any funeral mass or All Souls Day). Any of these sequences may still be said even in the Ordinary Form of the Mass. In each of these hymns, one can sense the medieval joy over the truths of the faith expressed in sublime poetry.