Does elasticity vary along the Demand/Supply curve or is it a summary statistic of a curve? I know that the Demand curve has elastic and inelastic regions but when we say horizontal demand/supply curves are elastic or vertical curves are inelastic, in that case, does the elasticity still vary along the curve?

More specifically, one of my lectures gives Elasticities of various items such as 0.09 for Gasoline and 0.8 for soda drinks. Are these values just taken on a specific point? If yes, how can they be used to describe the complete curve of a specific item.
Yes, elasticity varies along a curve. No, elasticity is not a summary statistic of a curve.

Definition. The elasticity of demand, $Q_D$, with respect to price, $P$, (or price elasticity of demand) is defined as this number: $$\frac{\partial Q_D}{\partial P} \div \frac{Q_D}{P}.$$



Example. Suppose $Q_D = 100 - P$. Then the price elasticity of demand (PED) is $$\frac{\partial Q_D}{\partial P} \div \frac{Q_D}{P}=-1\div\frac{Q_D}{P}=-\frac{P}{Q_D}=-\frac{P}{100-P}.$$




  More specifically, one of my lectures gives Elasticities of various items such as 0.09 for Gasoline and 0.8 for soda drinks. Are these values just taken on a specific point?


Yes. Elasticity estimates are usually point estimates.

Note though that sometimes they're some sort of a mean or average estimate. (In such cases anyone reporting such estimates should be very careful about how exactly these estimates were defined.)


  If yes, how can they be used to describe the complete curve of a specific item.


You're right. They don't describe the "complete curve".



Note. With soda, prices probably don't fluctuate too much. So elasticity estimates are probably reliable. 

In contrast, with gasoline (or more generally energy), we have to be more careful. Indeed, there's an extensive literature on this topic with estimates all over the place.

Here for example is a table from a recent meta-analysis (Labandeira, Labeaga, and López-Otero, 2017) where ranges of elasticities from various studies are reported: