Born in Vilamarxant, Valencian Community, Albiol started playing football with two modest clubs in his native region, moving to local giants Valencia CF before celebrating his 10th birthday. He made his first-team debut on 24 September 2003 in a UEFA Cup first-round tie against AIK Fotboll, at 18 years and 20 days, but spent the first year still registered with the B-side.

In August 2004, whilst travelling to sign a deal to join Getafe CF on loan, Albiol was involved in a serious car accident, being put in intensive care. He managed to recover completely and, after reappearing in January, played a very important part in the Madrid side surviving the relegation battle: he made his La Liga debut on 15 January 2005 in a 1–1 home draw with Atlético Madrid, and scored against another club from the capital, Real Madrid, in a 2–1 home triumph two months later.

Albiol returned the following season to the Mestalla Stadium, and quickly established himself as first-choice due to his versatility. In the opening game of the 2006–07’s league campaign he scored the winner (his only goal of the season) against Real Betis in a 2–1 home win. He also netted from long distance against Olympiacos F.C. on 12 September 2006, in that year’s UEFA Champions League, which Valencia won 4–2 away; during his four-season spell with the first team, his smallest output consisted of 29 games, and he helped the Che to the 2007–08 edition of the Copa del Rey.

After having represented Spain at under-21 level, Albiol first appeared for the senior squad on 13 October 2007 in an UEFA Euro 2008 qualifier against Denmark, a 3–1 away win. In the competition’s final stages, he appeared twice for the eventual continental champions, against Sweden (subbing in for the injured Carles Puyol) and Greece.

Albiol fathered two daughters with his wife Alicia. In 2008, with Guillermo Franco and Marcos Senna, he founded Evangélico FC, an organisation consisting of 140 athletes and 16 coaches which sought to promote Christian values among young athletes in Spain.

On 25 June 2009, Albiol joined Real Madrid for a fee thought to be in the region of €15 million, becoming the first Spanish player signed by Florentino Pérez upon his return to the presidency. On 2 July, he was officially presented at the Santiago Bernabéu, receiving the number 18 jersey from Rubén de la Red who was absent for the season due to unresolved heart problems.

Albiol scored his first goal for the Merengues on 8 December 2009, in a 3–1 Champions League win at Olympique de Marseille. He was an undisputed starter during the league campaign, mainly due to Pepe’s serious knee injury.

In 2010–11, Albiol was relegated to the bench as Real Madrid brought in another player for his position, Ricardo Carvalho (a compatriot of newly appointed manager José Mourinho), only appearing in the league through injury or suspension to teammates (Pepe again missed several matches due to physical problems). He did start, however, in the club’s domestic cup campaign, helping it reach the final. On 26 January 2011, in the semi-final first leg against Sevilla FC (1–0 away win), he cleared a Luís Fabiano shot just before it crossed the goal line; on 16 April, he was sent off for fouling former Valencia teammate David Villa inside the box during El Clásico, which resulted in the first goal in a 1–1 home draw, going on to miss the Spanish Cup final four days later – through suspension – a 1–0 win against the same opponent.

On 6 August 2012, Albiol renewed his contract with Real Madrid until June 2017. On 27 November, in his first game as team captain, at home against CD Alcoyano for the Spanish Cup, he had to be stretchered off after only one minute, being sidelined for approximately one month with an ankle injury.

Albiol scored his second official goal for Real Madrid on 8 May 2013, heading from a corner kick for his team’s first in an eventual 6–2 home win against Málaga CF.

On 21 July 2013, Albiol was sold to S.S.C. Napoli in Italy for a reported €12 million, signing a four-year contract with the Rafael Benítez-led team. He made his Serie A debut on 25 August, in a 3–0 home win over Bologna FC 1909.

Albiol scored his first goal for his new team on 25 January 2014, an 88th-minute equaliser in a 1–1 home draw against A.C. ChievoVerona. On 3 May, he played the entire final of the Coppa Italia, helping to a 3–1 success over ACF Fiorentina.

On 4 July 2019, the 33-year-old Albiol returned to Spain and agreed to a three-year deal at Villarreal CF. He made his debut on 17 August, in a 4–4 home draw to Granada CF.