The embodiments for which an exclusive property or priviledge is claimedare defined as follows: are defined as follows:1. A hand cleaning device, comprising:a housing including an inner cleaning compartment with an upper side wall portion and a lower side wall portion spaced from the upper side wall portion, the upper and lower side wall portions dividing the inner cleaning compartment into first and second cleaning chambers, the first and second cleaning chambers being in fluid communication with each other in the space between the upper and lower side wall portions;first and second access openings formed in the housing and communicating with the first and second cleaning chambers, respectively, each of the access openings being sized to receive a user's hand;a reservoir for storing a supply of cleaning fluid;at least one nozzle positioned in each of the cleaning chambers for directing cleaning fluid toward the user's hand;a pump fluidly connected between the reservoir and the nozzles for transferring cleaning fluid from the reservoir to the nozzles under pressure;a power supply connected to the pump for operating the pump; anda switch electrically connected between the power supply and the pump, the switch being operable to actuate the pump when a user's hand is positioned in one of the cleaning chambers.2. A hand cleaning device according to claim 1, wherein the switch is located in the cleaning compartment and is manually operable by the user.3. A hand cleaning device according to claim 2, wherein the switch is located in the space between the upper and lower side walls portions.4. A hand cleaning device according to claim 1, wherein the switch includes at least one proximity sensor located within the cleaning compartment for determining the presence of a hand within the cleaning compartment.5. A hand cleaning device according to claim 4, wherein the proximity sensor is located in the space between the upper and lower side wall portions.6. A hand cleaning device according to claim 4, wherein the at least one proximity sensor comprises a proximity sensor located in each cleaning chamber for detecting the presence of a hand within at least one of the cleaning chambers.7. A hand cleaning device according to claim 1, wherein the at least one nozzle comprises a plurality of nozzles positioned in each cleaning chamber.8. A hand cleaning device according to claim 1, wherein the at least one nozzle comprises a pair of nozzles positioned in each cleaning chamber, the nozzles of each pair of nozzles being oriented in opposing relationship such that a tip of each nozzle is directed toward a centerline of its respective access opening.9. A hand cleaning device according to claim 1, wherein the at least one nozzle comprises three nozzles positioned in each cleaning chamber, the three nozzles in each cleaning chamber oriented at approximately 120.degree. intervals around an imaginary centerline passing through its respective access opening, with a tip of each nozzle directed radially toward its respective centerline.10. A hand cleaning device according to claim 9, wherein at least one of the nozzles in each cleaning chamber is located on at least one of the upper and lower side wall portions.11. A hand cleaning device according to claim 10, wherein two of the nozzles in each cleaning chamber are located on the upper and lower side wall portions.12. A hand cleaning device according to claim 9, wherein the nozzles in each cleaning chamber are located in a plane transverse to their respective imaginary centerline.13. A hand cleaning device according to claim 1, wherein the at least one nozzle comprises four nozzles positioned in each cleaning chamber, the four nozzles in each cleaning chamber being oriented at approximately 90.degree. intervals around an imaginary centerline passing through its respective access opening, with a tip of each nozzle directed radially toward its respective centerline.14. A hand cleaning device according to claim 13, wherein two of the nozzles in each cleaning chamber are located on the upper and lower side wall portions.15. A hand cleaning device according to claim 13, wherein the nozzles in each cleaning chamber are located in a plane transverse to their respective imaginary centerline.16. A hand cleaning device according to claim 1, wherein the at least one nozzle is an atomizer nozzle for spraying a relatively fine mist of cleaning solution onto a user's hand.17. A hand cleaning device according to claim 16, wherein the cleaning fluid has a relatively quick evaporation rate.18. A hand cleaning device according to claim 1, and further comprising a door with a plurality of flexible flaps mounted in each access opening for preventing the escape of cleaning fluid out of the cleaning compartment.19. A hand cleaning device, comprising:a housing including an inner cleaning compartment having an upper side wall portion and a lower side wall portion spaced from the upper side wall portion, the upper and lower side wall portions dividing the inner cleaning compartment into first and second cleaning chambers, the first and second cleaning chambers being in fluid communication with each other in the space between the upper and lower side wall portions;first and second access openings formed in the housing and communicating with the first and second cleaning chambers, respectively, each access opening being sized to receive a user's hand;a reservoir for storing a supply of cleaning fluid;at least one nozzle positioned in each cleaning chamber for directing cleaning fluid toward the user's hand;a pump fluidly connected between the reservoir and the nozzles; anda power supply connected to the pump for operating the pump to thereby transfer cleaning fluid from the reservoir to the nozzles under pressure.20. A hand cleaning device according to claim 19, wherein the plurality of nozzles comprises a pair of nozzles positioned in each cleaning chamber, the nozzles of each pair of nozzles being oriented in opposing relationship such that a tip of each nozzle is directed toward a centerline of its respective access opening.21. A hand cleaning device according to claim 19, wherein the plurality of nozzles comprises three nozzles positioned in each cleaning chamber, the three nozzles in each cleaning chamber being oriented at approximately 120.degree. intervals around an imaginary centerline passing through its respective access opening, with a tip of each nozzle directed radially toward its respective centerline.22. A hand cleaning device according to claim 19, wherein the plurality of nozzles comprises four nozzles positioned in each cleaning chamber, the four nozzles in each cleaning chamber being oriented at approximately 90.degree. intervals around an imaginary centerline passing through its respective access opening, with a tip of each nozzle directed radially toward its respective centerline.23. A hand cleaning device according to claim 19, and further comprising a proximity switch located within the cleaning compartment for actuating the pump only when a user's hand is positioned in one of the cleaning chambers.24. A hand cleaning device according to claim 19, wherein the plurality of nozzles are atomizer nozzles for spraying a relatively fine mist of cleaning solution onto a user's hand.25. A hand cleaning device according to claim 24, wherein the cleaning fluid has a relatively quick evaporation rate.