Educated at Gresham's School between 1988 and 1993, Firman came from a motorsport family: his father Ralph Sr. co-founded Van Diemen, a prolific junior formula constructor, while his sister Natasha also pursued a career in racing. Firman competed under an Irish racing licence for much of his career. He is married to Aldís Kristín Árnadóttir, an Icelandic UK-educated lawyer.
Firman entered the British Formula 3 Championship in 1995 and was competitive immediately, leading much of the season before losing the title at the final round to Oliver Gavin. He returned in 1996 and captured the championship at his second attempt, claiming the title he had so narrowly missed the year before. That same year, he won the prestigious Macau Grand Prix under dramatic circumstances: in the two-leg race, Firman finished ahead of German Formula 3 champion Jarno Trulli at the end of round one, but in round two Trulli overtook him on the final lap after Firman broke his front wing. Firman then crashed at the hairpin corner, blocking the track. The race was stopped and results declared at the end of the previous lap, at which point Firman held the lead — giving him the victory despite not finishing the second leg.
Firman moved to Japan after his Formula 3 success, embarking on a career that spanned more than a decade in Japanese motorsport. He competed in Formula Nippon, the country's premier single-seater series, and won the championship in 2002 — the high point of his open-wheel career. He also competed extensively in Super GT, the Japanese touring car series for high-performance GT machines. He was runner-up in the Super GT Series in 2002, 2005, and 2009, and took overall victory in 2007 alongside co-driver Daisuke Ito with the ARTA team, co-owned by former Formula One driver Aguri Suzuki. The 2007 Super GT title was clinched before the final round of the season, a first in the series' history.
Firman secured a Formula One seat for the 2003 season with the Jordan team, partnering Giancarlo Fisichella. He participated in 14 Grands Prix, making his debut at the 2003 Australian Grand Prix. His campaign produced one championship point, scored at the 2003 Spanish Grand Prix — a result that was also the first point for a Republic of Ireland driver in Formula One since Derek Daly in 1982. His season was disrupted by a serious crash during practice for the 2003 Hungarian Grand Prix, which forced him to sit out that race and the next; Zsolt Baumgartner replaced him in the Jordan during his absence. Firman did not return to Formula One after the 2003 season.
In November 2003, he drove a Jordan-Ford EJ13 at the Macau Grand Prix's 50th anniversary celebrations — the first time a contemporary Formula One car had run on the tight Guia Circuit. He lapped in 1:59.4, some 13 seconds quicker than the Formula 3 poleman on the same day.
In addition to his Japanese career, Firman competed in the 24 Hours of Le Mans and participated in the A1 Grand Prix series, representing Team Ireland after initially being considered for the Great Britain seat. He retired from racing in 2013 and subsequently ran a British engineering company. His racing career reflects a driver who found his natural habitat in Japanese motorsport rather than European Formula One: his Formula Nippon championship win and dual Super GT titles stand as his most significant achievements.