Albers began karting at a young age and won the Dutch National kart championship in 1997, the same year he claimed the Formula Ford 1800 title in both the Netherlands and Belgium. He moved to the German Formula Three series in 1998 and won the championship in 1999 with six victories and ten pole positions. In 2000 he raced in the International Formula 3000 championship alongside future world championship contender Mark Webber without scoring points, but team owner Paul Stoddart maintained confidence in him.
From 2001, Albers competed in the Deutsche Tourenwagen Masters for Mercedes-Benz, graduating from a privately run entry to the works AMG squad in 2003. That season he finished runner-up in the championship to Bernd Schneider, scoring four victories and contesting the title until the final round. He again challenged at the front in 2004, finishing third. He remains the Netherlands' most successful driver in the DTM series.
While continuing in the DTM, Albers served as a test and reserve driver for Minardi. He was selected to race in the 2005 Formula One season, and at the 2005 United States Grand Prix — contested by only six drivers after the Michelin-equipped teams withdrew over tyre safety concerns — he claimed his first and only championship points with a fifth-place finish.
Albers was confirmed as one of Midland's drivers for 2006, partnering Tiago Monteiro in the renamed Jordan outfit. He made a strong start to the season, out-pacing his teammate in the early rounds. At the San Marino Grand Prix he was struck by Yuji Ide and sent into a dramatic series of rolls, emerging unharmed. Ide subsequently had his FIA Super Licence revoked for dangerous driving.
The Midland team was acquired by Spyker Cars and rebranded as Spyker F1 for 2007, and Albers was retained alongside new signing Adrian Sutil. He was outperformed by his rookie teammate throughout the first half of the season. At the French Grand Prix in Magny-Cours he drove away from his pit box with part of the fuel rig still attached, resulting in a dangerous situation and a five-thousand-euro penalty. On 10 July 2007, following the British Grand Prix, Spyker terminated his contract citing insufficient sponsorship funds that would have compromised the team's development programme. Markus Winkelhock replaced him at the European Grand Prix, with Sakon Yamamoto taking the seat for the remainder of the year.
When Tony Fernandes sold the Caterham F1 Team to a consortium of Swiss and Middle Eastern investors in July 2014, Cyril Abiteboul departed as team principal and Albers was installed in the role. He managed the team through the latter part of the 2014 season until Caterham collapsed financially following the Abu Dhabi Grand Prix.
Albers returned to the DTM in 2008 as a driver for the Audi Futurecom TME team, racing a 2006-specification Audi A4 alongside Katherine Legge. He also entered sportscar racing, co-driving an Audi R10 TDI at the American Le Mans Series finale at Laguna Seca in 2008 with Emanuele Pirro, finishing second, and competing in the Le Mans Series for the Kolles-run customer Audi squad in 2009.
Albers established a second career as a television analyst, joining the Dutch broadcaster Viaplay from 2022 onwards as a permanent member of their Formula One commentary and analysis team alongside former drivers Giedo van der Garde, Tom Coronel, and Ho-Pin Tung.