Market Value is the value of the property in today's market - based on what comparable properties have sold for recently. This is the most important value if you wish to purchase and are financing (not paying ALL cash) since this will be closest to appraised value.
Assessed Value is just the value used to compute property taxes. Has no relevance for purchase price.
Listing Price is just the price the seller has decided to list at. It may be too high or too low in comparison to appraised value.
If you are going to make an offer on a property, the best thing to do is have your Realtor do a Comparative Market Analysis (CMA) on the property using SOLD comps within a 1 mile radius of the property (the closer, the better) that have SOLD within the last 3-6 months. This will give you current market value and this is what you should base your offer on, not on list price or assessed value.