Who is entitled to be wali, and in what order?

The order of wali by lineage (wali nasab) in the Shafi'i school is: father, paternal grandfather (and upwards), full brother, paternal half-brother, nephew (son of the full brother, then of the paternal half-brother), paternal uncle (full, then paternal half), and male cousin (son of that uncle).
The nearer wali (aqrab) must take precedence; a more distant wali (ab'ad) cannot conduct the marriage while a qualified aqrab wali exists. A maternal half-brother and a maternal uncle are not wali nasab. If there is no wali nasab at all, the wali hakim takes over. See Marriage wali.