Dear Marie:

Vespers is the evening prayer of the Divine Office (Liturgy of the Hours).  Fr. Hardon's Dictionary explains:

[The Divine Office is a] group of psalms, hymns, prayers, biblical and spiritual readings formulated by the Church for chant or recitation at stated times every day. Its origins go back to apostolic times, when it consisted almost entirely of psalms and readings from the Scriptures. Priests are obliged to say the full daily office, and religious who are not priests are obligated according to their rule of life.

The offices since the post-Vatican II revisions are:

The times for the offices are traditionally about every 3 hours, but that does not have to be firmly set as long each hour is recited around the proper times of day for which they are intended. So if one misses morning prayer because they did not get up until 8am, they miss it and cannot "make it up" later in the day.

Office of Readings (Vigils)... recited around 3am or can be said at any time of day, even the night before

Morning Prayer (Lauds)... to be said in morning (e.g., 6am)
Mid-Morning Prayer (Tierce)... to be said in mid-morning (e.g., 9am)
Mid-Day Prayer (Sext)... to be said around noontime
Mid-Afternoon Prayer (None)... to be said in Mid-Afternoon (e.g., 3pm)
Evening Prayer (Vespers)... to be said in the evening (e.g., 6pm)
Night Prayer (Compline)... to be said before retiring for the night, whenever that is

The purpose of these "hours" is to sanctify the various times of the day and to obey Christ's call to "always be in prayer". Somewhere in the world it is always time for Divine Office, so the Church is literally praying always, 24 hours a day.

Benediction is when the exposed Blessed Sacrament is reposed to the Tabernacle. As explained by Fr. William Saunders in his article, Benediction of the Blessed Sacrament:

The ritual for exposition and benediction as presented most recently by the Sacred Congregation for Divine Worship (1973) basically follows this ritual: The priest places the Blessed Sacrament in a monstrance or ostensorium on the altar for adoration. (A ciborium containing the Blessed Sacrament may also be used, but the monstrance allows one to view the holy Eucharist.) At this time, a hymn of praise (such as O Salutaris Hostia) is sung as the priest incenses the Blessed Sacrament.

During the period of adoration, the faithful may pray in quiet and foster a deeper spiritual communion with the Lord. However, the adoration period should also include prayers, such as a novena or Liturgy of the Hours, and readings from sacred Scripture accompanied perhaps by a homily or exhortation to increase the understanding of the eucharistic mystery.

At the end of the period of adoration, the priest again incenses the Blessed Sacrament as a hymn of praise is sung (such as Tantum Ergo), and then blesses the congregation with the Blessed Sacrament, making the sign of the cross. After the blessing, the priest reposes the Blessed Sacrament in the tabernacle.

God Bless,
Bro. Ignatius Mary