The number of contracts remaining open at the end of the trading session is the open interest. The open interest increases if more positions are opened or fewer ones closed compared to the open ones; it decreases if more positions are closed than those opened.
For, e.g., if you purchase as a part of your trading strategy but, do not sell, this kind of trade is said to be open because it is subject to price variation. Also, if you sell and no identical purchase is made, similar is the case.
Now comes the question of the relation to spot prices and futures. Let us look into the two types of assets, the investment assets and the consumption asset. Investment asset is a term for an asset held by a number of investors for investment purposes, and the cost of carry includes the income from the asset subtracted from the storage cost (if there) added with the interest paid as a part of financing the asset.
A consumption asset, as the name says, is held for consumption. Related to consumption assets is the term convenience yield which is a measure of lost benefits derived from holding futures and not having goods in physical stock.
Now, for a consumption asset, the difference between the cost of carry and the income or yield determines the future prices of the asset. Thus is the relationship between future markets’ prices and spot prices. Though as expiration is approached, the two prices differ lesser and lesser as future prices merge with the spot ones.