In both formal and informal ways.
F
Informally, we "offer it up" by simply asking God in our own words to use a suffering as it occurs; we often do this for specific intentions (ex., "Use this pain, Lord, for the salvation of my brother..."). We might follow the example of the young St. Thérèse of Lisieux and make use of Sacrifice Beads, or the extraordinary among us might make the Heroic Act of Charity for the souls in Purgatory.
It's quite a discipline to react to suffering this way! In mental or physical pain? Drop something on your toe? Putting up with a co-worker who is making your life a living Hell? Enduring the constant ache of arthritis? Standing in line at the grocery and hating every minute of it? Spill the milk? Accept these things in peace, and ask God to use them for the good of the Church or for a more specific
You'll find that it is not uncommon to hear one Catholic tell another who is suffering to "offer it up" as a way of dealing with his suffering. It should be remembered, though, that while it is most definitely good to tell someone to "offer it up," it is also easy -- and that we are called, too, to comfort those who are suffering, to feed the hungry, to give drink to the thirsty, to care for the sick, etc. Telling someone to offer it up without also helping him to deal with the temporal and emotional effects of whatever they are going through is not the fully Christian thing to do. Even Our Lord was helped while carrying His Cross: St. Veronica wiped the sweat and Blood from His Holy Face, and St. Simon of Cyrene helped Him bear the Cross itself.
And always help the suffering to retain (or regain) Hope that his suffering is not in vain. Assure him that he will partake of "the consolation":