Born in La Pobla Llarga, Valencian Community, Albelda started his career as a central defender with local UD Alzira and, after having served two separate loans with Villarreal CF, also in the region, returned for good to first professional club Valencia CF in the 1999–2000 season, playing a major role in a squad that was crowned La Liga champions in 2002 and 2004, adding to that the 2004 UEFA Cup;[5] since summer 2001, with Gaizka Mendieta's departure, he was also awarded team captaincy.

[6] Albelda, along with teammates Santiago Cañizares and Miguel Ángel Angulo was axed from the squad on 20 December 2007 by recently arrived coach Ronald Koeman;[7] in the captain's case, he had ironically just renewed his link with the Che until 2011.

[8] In response, on 2 January 2008, Albelda's lawyer claimed that Valencia had ignored his client's plea for an amicable end to the player's contract, with legal recourse the only possible avenue left.

He returned to action in the second half of the 6–0 away loss against FC Barcelona on 4 May,[10] and received his first post-reinstatement start in a 5–1 victory in the derby away at Levante UD the following week, as the side mathematically retained top-flight status.

Albelda also won a silver medal at the 2000 Summer Olympics in Sydney,[2] and represented his nation at the 1997 FIFA World Youth Championship.