The Altar is the structure on which the sacrifice of the Mass is offered. In unusual circumstances such in mission lands or on the battlefield any usable surface will do, but usually Mass is offered on a proper altar in a church.
“I saw under the altar the souls of them that were slain for the word of God.” (Rev. 6:9.)
In the early centuries of the Church, Mass was sometimes celebrated over the graves of the martyrs, in the Catacombs. When the Church came out of the Catacombs the tradition of placing relics of the saints in or under the altar became universal.
In recent centuries they were often sealed in a removable altar stone rather than the altar itself. When the priest kisses the altar it is this place he kisses, venerating not only the altar itself (which is a representation of both Christ and the Cross) but also the saints whose relics are present.
“We beseech You, O Lord, by the merits of Your Saints whose relics lie here, and of all the Saints, deign in your mercy to pardon me all my sins. Amen.” (Said in the traditional mass just before the Kyrie.)
Upon the altar there are three cloths these represent the grave cloths that wrapped the body of our lord. There are also two smaller candles on each side of the altar (or on the altarpiece behind it); these are used at Low Mass. Behind these are six (or four) larger candles used at High Mass. Candles must be at least 51% beeswax.
In the center of the altar there is a crucifix. (There may or may not be a tabernacle in the center of the altar, it is to be covered with a veil.) The altar missal is on its stand; if the book is closed the page edge always is pointed toward the center of the altar. (I don’t know what the significance of this is.)
At the center of the altar and at each side there are altar cards containing often-used parts of the Mass for the priest's use. (Printable altar cards can be found here: Calefactory, or here: Rosary.)
“Almighty God, we pray that your angel may take this sacrifice to your altar in heaven. Then, as we receive from this altar the sacred Body and Blood of your Son, let us be filled with every grace and blessing.” (Said after the consecration at Mass.)