I don't know Greek but I checked some other translations and that helped a lot. NASB (the most literal* of all English translations) refers to that band of soldiers as a "Roman cohort"; NLV describes them as "contingent of Roman soldiers". Here are some other translations.
I read up in Nelsons New Illustrated Dictionary and found this on Cohort/Regiment:
"One of ten divisions of ancient Roman Legion. The traditional Roman legion consisted of 6,000 soldiers. A regiment or cohort, consisted of about 600 men, although this number varied. The Book of Acts mentions the Italian Regiment (Acts 10:1; band, KJV; cohort, NRSV, REB, NASB) and the Augustan Regiment (Acts 27:1; band, KJV; Imperial Regiment, NIV)"
So we know that the band of soldiers were roman for sure, most likely the High Priests wanted to secure Jesus because they had been waiting so long to finally capture him in a situation that wouldn't stir up any trouble, so they could quickly kill Jesus and nobody would remember him. Ironically, things didn't go as planned for them because God's plan was that through Jesus' death and ressurection God would be glorified and our sins would be forgiven. The High Priests and Pharasies didn't realize that killing Jesus would make him even more famous and give them even more problems.
Obviously they didn't bring 600 men to arrest Jesus, I'm assuming that a selection of the Roman cohort was asked by the high priests probably decaalring to their (Roman) rulers that the situation required such extreme measures. At this time under Roman rule the arrests where performed by roman government and officials, including their army and special forces. In modern day terms it would be like taking the SWAT to arrest Jesus. You wouldn't take the entire team but a good amount becuase they new Jesus always had his disciples with him.
I hope that helps, my advice is to research it on Biblios.com, a great resource! Praise God that he used mans' hatred towards Jesus to put him on a cross where he took the wrath of our sins and waved victory over our bonds to sin!
* Here is a good chart to see the most literal to least literal of all English translations.