My provider said they can provide single / mulit mode Optical fiber
Apart from the length and cost different, what is the Adv/Disadv between them for our connection?
The advantages are always in the distance capabilities of the single mode fiber. You can reach much further on this, but the optics tend to be more expensive. If you are going a short distance (eg: 2km or less) multi-mode is the way. If you're going to go any further, or want to ever go any further, take the extra cost and know you can swap optics in the future to do gig, 10G and possibly more (in the future) with less pain.
Just to amplify Jared's very complete answer. The principle reason you would use multimode instead of single mode is reduced cost. If cost isn't an issue, single mode has more potential to be used in more applications. Even the longer range SM optics can be used for short range uses with inexpensive attenuators. Service Providers support both because their customers may only support one or the other. Deepak Jain AiNET