One concept of Apostolic Succession, sometimes called "tactile succession", is that bishops derive their position by being consecrated by other bishops, who were themselves consecrated by earlier bishops, and so on right back to the apostles, and so to Christ Himself. A valid bishop is someone in this chain. A valid priest is someone appointed by a valid bishop. A priest, who is not a bishop, cannot pass on his priestly status to others, whereas a bishop can pass on priestly status to others, and can also pass on episcopal status to others.
The weakest link in the Anglican chain of succession was seen as Matthew Parker appointed by Elizabeth I after the death of Queen Mary. Cranmer and other Protestant-minded bishops had been burned at the stake under Mary. The Catholic-minded bishops appointed by Mary resigned or were deprived and refused to consecrate Parker. Cardinal Pole, Archbishop of Canterbury between Cranmer and Parker, died the same day as Mary. By ancient tradition three bishops, at least, participate in the consecration of a new bishop.
Parker's consecration was delayed for months, but eventually 4 bishops were found: Barlow, Coverdale, Scory and Hodgskin. Of these Barlow and Hodgskin had been consecrated in the Roman rite in the time of Henry VIII; and Coverdale and Scory had been consecrated by Cranmer in his own rite under Edward VI (son of Henry, brother of Mary and Elizabeth). All Anglican bishops were, and are, "descended" from Parker, and through him from Cranmer and from the Roman Catholic Church of which the Church of England was once part.
Allegations that Parker had never been formally consecrated, but merely taken part in an unofficial ribald ceremony in a London pub, the Nag's Head , were made by Jesuits, though these are not accepted by historians, and have never been officially accepted by the Roman Catholic Church. There are also legal issues, but these have no bearing on validity.
So, why does the Roman Catholic Church regard Anglican orders as invalid?
In 1896 Pope Leo XIII issued a bull, Apostolicae Curae, in which he declared Anglican orders to be absolutely null and utterly void.
There were several reasons. One concerned the form, the form of words in the consecration rite, in the Edwardine Ordinal used from 1551 until 1662 (except for Mary's five-year reign when the Roman rite was restored, and the Cromwellian Period when bishops were abolished). Following the words "Receive the Holy Ghost" the word "bishop" was not mentioned, nor anything definite to say what a bishop is. Although this was fixed, so to speak, in the 1662 Book, Leo said the defect of form meant there were no valid consecrations in England for over 100 years and so the hierarchy had died out. Even if the form was acceptable after 1662, there were no valid bishops to use it. The Anglican line had died out.
Since the 1930s it has been standard practice for bishops from the Old Catholic churches to co-consecrate Church of England bishops. Old Catholic Churches split from Rome, mostly over Vatican One and Papal infallibility, and their bishops are accepted by Rome as valid. Most, perhaps all, Anglican bishops worldwide can now trace their "descent" from Old Catholic, as well as Anglican and Lutheran, bishops. Most Old Catholic Churches are members of the Utrecht Union, after the town of Utrecht in Holland. For this reason the participation of Old Catholic co-consecrators is known colloquially as the "Dutch Touch".
However the Dutch Touch does not deal with Leo's principle objection to Anglican orders which is not about form, but intention. Anglican ordinations of priests, and consecration of bishops, are invalid because there is no intention, in the Anglican churches, that they should be valid. An ordination using an Anglican rite, in an Anglican church, must be assumed to have the public intention of accomplishing what the Anglican church says it is intended to accomplish, and that is fundamentally not to make priests or bishops in the Catholic sense of those terms.
The Archbisops of Canterbury and York issued a response to the bull from Leo, and the Roman Catholic bishops of England and Wales issued a Vindication of the bull, in the form of a reply to Canterbury and York. The Vindication is perhaps clearer than the bull. It describes Catholic doctrine.
Priest and Sacrifice are correlative terms -with us at all events, and indeed with all nations, except in so far as your own Communion may be an exception. A priest is one who offers sacrifice; and as is the sacrifice, so is the priest. Since, then, our sacrifice is the Sacrifice of the Mass, our priest is one appointed and empowered to offer up that sacrifice; one, therefore, who has received from God the power, by means of the words of consecration, to cause the Body and Blood of Christ to become present under the appearance of bread and wine, and to offer them up sacrificially.
Having explained the Catholic understanding of the word "priest", it goes on to document the Church of England position based on her Articles of Religion, her liturgy and the statements of her reformers and prominent theologians. The Church of England rejects the Roman Catholic understanding of the sacrifice of the mass as a blasphemous fable.
The true Sacrifice and Priesthood ..... your Church has in the most stringent terms repudiated altogether.
So , although Anglicans use the words "priest" and "bishop" they do not mean what Catholics mean, but something else. When Anglicans "ordain a priest" they have no intention of doing what Catholics do when they "ordain a priest". That, rather than tactile succesion, is why the Roman Church deems Anglican orders invalid.