“Vespers” is Evening Prayer from the Liturgy of the Hours (also called the Divine Office or the Breviary), part of the official liturgical prayer of the Church. Priests and religious are bound to pray the Liturgy of the Hours several times each day. Lay faithful are not obligated but are welcome to pray these prayers as well. The prayers for the entire year are contained in a set of four volumes of books (the Breviary) — one volume for Advent/Christmas season, one volume for Lent/Easter season, and two volumes for Ordinary Time.
Each of the seven “hours” of the day includes a hymn, several selections from the Book of Psalms, other reading(s) from a different part of Scripture, and other prayers. The Book of Psalms is pretty much entirely covered in a four-week cycle if you pray all seven hours every day. Evening Prayer (Vespers) includes the Magnificat, Mary’s prayer of praise during her visit to Elizabeth that we read in the Gospel of Luke.
Sunday evening Vespers is considered the most important hour of the Divine Office each week, and sometimes on special occasions it is prayed with greater solemnity, e.g., singing rather than reciting the Psalms, using incense, etc. — hence the term “Solemn” Vespers.
I have attended Solemn Vespers on a couple of occasions, and it was very beautiful and prayerful. They will probably provide printed handouts so you can pray along. Enjoy it!