Xiphos (ξίφος). The most well-known type, a short two-sided sword used by hoplites as a secondary weapon.
Xyele (ξυήλη). A shorter infantry sword than the xiphos, used by the classical era Spartan hoplites. Basically a long dagger, there’s still debate on whether it was curved or not.
Kopis (κοπίς). A heavier single-edged cutting sword, usually employed by cavalry.
Makhaira (μάχαιρα). Generic name for a single-edged knife, scalpel or short sword.
Spatha or spathíon (σπάθη ή σπαθίον). A straight long sword of the late Antiquity, used by the Roman empire military as a replacement of the shorter gladius. Spatharios was an Eastern Roman title which literally means spatha-bearer.
Rhomphaia (ρομφαία). A long slightly curved blade used by the Thracians, similar to a falx.