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Neoclassical economics is an economic theory that argues for markets to be free. This means governments should generally not make rules about types of businesses, businesses' behaviour, who may make things, who may sell things, who may buy things, prices, quantities or types of things sold and bought. The theory argues that allowing individual actors (people or businesses) freedom creates better economic outcomes. These outcomes may be a higher average standard of living, higher wages, better average life-expectancies, and higher GDP.
Markets are an abstract idea: assumed to be all the 'actors' (businesses or people) selling one thing, service or type of thing or service, and all the 'actors' buying it.
Markets will 'reach equilibrium' if all the sellers who want to sell at or below a given price have sold to all the buyers who are willing to buy at or above a given price. the price is worked out in the market.
It may be easier to think about this in reverse: The market is not in equilibrium if people want to buy a haircut for ten (or more) dollars and someone is happy to sell the person a haircut for ten (or less) dollars, but for some reason this does not happen.
Neoclassical economists say this will not happen. Neo-Keynesians say it might, so the government could make the customer and the person selling the haircut happier by helping the customer somehow.
Neo-Keynesian economy is an alternative to Neoclassical economy. The major point of difference between neoclassical economics and neo-Keynesian economics is about whether 'markets' 'reach equilibrium'.