How is Union with Christ understood across Christianity?

How is Union with Christ understood [in Catholicism]?

The third stage/way/state of the spiritual life is the unitive.

It is called the state of "perfect charity", because souls who have reached that state are ever prompt in the exercise of charity by loving God habitually and by frequent and efficacious acts of that Divine virtue. It is called the "unitive" way because it is by love that the soul is united to God, and the more perfect the charity, the closer and more intimate is the union. Union with God is the principal study and endeavor of this state. It is of this union St. Paul speaks when he says: "He who is joined to the Lord, is one spirit." (1 Corinthians 6:17). Souls thus united to God are penetrated by the highest motives of the theological and moral virtues. In every circumstance of their lives the supernatural motive which ought to guide their actions is ever present to their mind, and the actions are performed under its inspiration with a force of will which makes their accomplishment easy and even delightful. These perfect souls are above all familiar with the doctrine and use of consolations and desolations. They are enlightened in the mysteries of the supernatural life, and they have experience of that truth proclaimed by St. Paul when he said: "We know that to them that love God, all things work together unto good, to such as, according to His purpose, are called to be saints." (Romans 8:28). 1

Aside from this habitual state of imitation of Christ, we could also say that one experiences union with Christ in other senses at different times, but someone more learned than me would have to enumerate them all.

We could certainly say we are united with Christ when we are members of His Mystical body (i.e. the Church):

The analogy borne by any society of men to an organism is sufficiently manifest. In every society the constituent individuals are united, as are also the members of a body, to effect a common end; while the parts they severally play correspond to the functions of the bodily organs. They form a moral unity. This, of course, is true of the Church, but the Church has also a unity of a higher order; it is not merely a moral but a mystical body. This truth, that the Church is the mystical body of Christ, all its members being guided and directed by Christ the head, is set forth by St. Paul in various passages, more especially in Ephesians 4:4-13 (cf. John 15:5-8). The doctrine may be summarized as follows:

• The members of the Church are bound together by a supernatural life communicated to them by Christ through the sacraments (John 15:5). Christ is the centre and source of life to Whom all are united, and Who endows each one with gifts fitting him for his position in the body (John 15:7-12). These graces, through which each is equipped for his work, form it into an organized whole, whose parts are knit together as though by a system of ligaments and joints (John 15:16; Colossians 2:19).
• Through them, too, the Church has its growth and increase, growing in extension as it spreads through the world, and intensively as the individual Christian develops in himself the likeness of Christ (John 15:13-15).
• In virtue of this union the Church is the fulness or complement (pleroma) of Christ (Ephesians 1:23). It forms one whole with Him; and the Apostle even speaks of the Church as "Christ" (1 Corinthians 12:12).
• This union between head and members is conserved and nourished by the Holy Eucharist. Through this sacrament our incorporation into the Body of Christ is alike outwardly symbolized and inwardly actualized; "We being many are one bread, one body; for we all partake of the one bread" (1 Corinthians 10:17). 2

Or when we are simply in a state of grace:

Union with God belongs substantially to all souls in a state of grace, but it is in a special manner the distinguishing characteristic of those in the unitive way or in the state of the perfect. 1

The friendship of God is consequently, one of the most excellent of the effects of grace... According to the Scriptural concept (Wisdom 7:14; John 15:15) this friendship resembles a mystical matrimonial union between the soul and its Divine spouse (Matthew 9:15; Revelation 19:7). 3