Apostolic Succession has nothing to do with the Pope. It means that a Bishop was ordained by another Bishop, and he was ordained by a Bishop, and so on until you get back to an Apostle. The Greek Orthodox are one of several Eastern Catholic communions which have Apostolic succession (and valid Holy Orders), as well as the Old Catholics and the Polish National Catholic Church.
Apostolic succession is how the Church transmits its authority. We see an example of this when Paul ordains Timoty and Titus, and instructs each of them to ordain others, and gives them a list of qualifications for Bishops.
Apostolic Succession is required for Ordination. If a person is ordained as a Bishop, he may transmit his Orders to others. Most people are ordained as priests. A priest may not ordain, but he has the authority to conduct any other Sacrament (in the case of Confirmation, as an extraordinary minister). This includes Eucharist.
Apostolic succession alone is not sufficient to transmit Orders. Like all Sacraments, Holy Orders also requires valid subject, form, matter, and intent. The Anglican Church originally possessed valid Orders, but lost them due to a theological shift which invalidated their intent (and probably form). By the time this was corrected by the Oxford movement, the Anglicans had no valid Bishops left. Anglicans had (and still have) valid succession, but invalid Orders.
We know that our Apostolic succession is valid beyond all possible doubt. We don’t know the lineage of any Bishop (meaning we don’t know the names or identities of all of that Bishop’s forebears), but we don’t need to. We know this from math. I explain how in post #7 of
this thread.