So is this like appointing a pope.
It's similar, yes. Anglican churches, like the Catholic church, are organized into dioceses, with a bishop in charge of each diocese. In the US, there are about 2-3 dioceses per state. Wales has 6 dioceses. Anglican churches in other countries are similarly organized regionally.
In many countries there are larger regions that contain multiple dioceses. The person in charge of these larger regions is usually called an archbishop. In the case being discussed in the OP, Wales has an archbishop who is the chief pastor overseeing the entirety of the (Anglican) Church in Wales, and that's the office we're discussing.
"In charge of", here, is a combination of administration and pastoring, as I mentioned earlier. We've come a long way from feudal Europe, so bishops and archbishops aren't like kings. But someone has to set policy and to make sure rules are generally followed and to resolve disputes and to make sure the diocesan budget is in order, and so forth, and that's part of the job of the bishops and their staff. There's also a pastoral role, giving spiritual guidance to the diocese and celebrating sacraments like Confirmation and Ordination.
There are two large differences between the Anglican Archbishop of Wales and the Pope: 1) The Pope is the chief bishop of the entire Catholic Church, whereas the Archbishop of Wales oversees a much smaller region. 2) Anglicans don't have any sense of infallibility of their bishops and archbishops. Bishops do their best to teach and minister faithfully, but we understand that they are flawed human beings like everybody else.
