When I work with multiple ISPs, they refer a connection type as a "10 Gbps wave connection". What does it really mean? I can understand 10 Gbps fiber but what does "wave" meas?
Most likely they mean a wavelength on a DWDM path. With DWDM, a fiber operator can use one single pair of fibers to carry a number of different colors (wavelengths) of light using multiplexing. Each of these colors can then be used to transmit its own signal, for example 10Gbps ethernet. This way, they are able to use a fiberpair more efficiently.
A wavelength service is muxed onto a DWDM system, such that you get a full 10Gbps pipe non-blocking with a fixed latency and jitter (except in case of protection switching if the provider has a failure!). You're geting either a full color of light on a fiber or at least a 10gbps slot of a color of light which is muxed in a TDM fashion (that is, withOUT statistical multiplexing). In modern networks this is probably done with OTN, in the past it would've been SONET (OC-192 for example).
So this is in contrast to an MPLS or packet-based service where the provider would provision your 10Gbps on routers/switches and you'd share some statistically-muxed packet queues with other customers. This is usually cheaper because the provider can over-provision more heavily.
Whether you care depends on your application for the cuircuit and the quality of the provider.
Wavelength (colors) used for the transmission are longer than visible light and lie in infrared band. The standard wavelengths are 850nm, 1310nm and 1550nm depending on the type of fiber. While plastic optical fiber (POF) and multimode fiber transmission use shorter wavelengths between 650nm-850nm and 850nm-1310nm, respectively, single mode fiber transmission operates at 1260nm-1675nm. The wavelength range is lower confined by the fiber cut-off wavelength and upper restricted by bending and SiO2 absorption losses
Wave is the name of a company which offer carrier transport service and IP transit, on the US west coast.