While working in retail industry, I remember way back in 2007-2008 I heard of a buzz world Multi-Channel, and since then have been hearing all sort of terms Cross-channel and omni-channel. Interestingly enough all these terms cross boundaries, which is easy enough to confuse naive users.
Here is my take on these three terms to make them hopefully little more clearer.
Multi-channel is the experience of a customer who shops using the different channels made available by a company, such as brick-and-mortar stores, catalogues, web, mobile/tablet application, TV commercials, and call centres. This is not a new concept, we have had consumers shopping via desktops, tablets etc. However the emphasis is growing number of channels and how the technology should be flexible to accommodate these new channels (e.g. a possible smart car with shopping capabilities).
Cross-channel suggests of a user journey where user starts on one channel/device and moves to other device/channel and then to another channel. Classic example is a user does research for product via mobile/tablet while travelling, then uses Call centre to get some queries answered and eventually does a purchase using web channel (to take web exclusive promotions).
Omni-channel suggests simultaneous use of two or more channels, like using a mobile phone while in-store, or web while calling call centre, a tablet while watching TV. The term is also used to describe the consistency between different channels that facilitates and streamlines customer interactions.