A normal futures curve is when the futures prices are higher than the underlying in the spot market and are continuously increasing. An inverted curve is when the futures prices are lower than the underlying in the spot market and are decreasing. For instruments that are not seasonal in nature, normal futures curve is the natural state of pricing. A normal market expects that the futures prices will be always higher than the underlying spot price. As the expiry dates are further out to the future, markets expect the futures prices to go higher.
Normal futures curve
Futures curves are drawn by plotting the price of futures contracts at different maturities in a single graph.

Above example is plotted with data from CME on bitcoin futures settlement prices. On the 1st of April, all futures prices on bitcoin are higher than the spot price of bitcoin. Furthermore, all futures with far out expiry dates have a higher price than the ones that are nearer to expiration.
Futures further out are priced higher because of a simple reason, interest rate. Let’s take the example above. If you bought a 1/100th of a bitcoin on 1st April, that would have cost you 590.95 USD. There is no cost for you to hold the bitcoin; there are no warehouse charges, there are no account minimum charges etc. Let’s now suppose at the same time, your friend decides to keep their money and invest later.
At this point, you and your friend can agree or disagree on whether the bitcoin price will go up or down in the future. But that is a speculation. What is undisputed is that your friend will earn some interest if he banks it. Now if you agree to sell your bitcoins to your friend, you would at least ask for that interest rate return. Since interest rate is a positive number, the futures price is always expected to be higher than the spot price.
Inverted futures curve
An inverted curve is when the futures prices are lower than the underlying in the spot market and are decreasing.

In the example above we can see that spot price of WTI is 61.49, but all futures prices are below that price. Furthermore, prices for futures expiring further away are lower than the prices of the nearer months. The reason for normal futures curve can be explained well. But there is no one reason for inverted curves.
Inversion can occur naturally with seasonal commodities. Let’s suppose the underlying is a grain which is due harvest in June. If you consider the May and June futures, the May future will be higher because the supply is from the previous harvest. However in June, new harvest will come in and there will be a fresh supply. Therefore the prices will drop.
Negative interest rates will also contribute to inversion. If we take the bitcoin example, the money in bank will mean less valuable in the future. Therefore you do not have an opportunity cost by investing in bitcoins now. But other investors who want to buy the future know that you do not have a choice of holding the money in bank either. This exerts a downward pressure on the futures prices until it reaches an equilibrium.
In the case of WTI however, the reason is largely speculation. While as of now there is a demand for crude oil and the price is around 61 USD, market believes that eventually the production will increase and the prices will come down.
Normal futures/inverted curves vs contango/backwardation
There is sometimes confusion over normal futures/inverted curves and contango/backwardation. The normal futures and inversion curves are identified by plotting the prices of all futures contracts with different maturities. If the prices of the futures expiring later are higher than the prices of the futures expiring nearer, then the futures curve is normal. If the futures expiring later are lower than the prices of futures expiring nearer, then the futures curve is inverted.
Contango and backwardation refer to the price of a given futures contract. Let’s take a bitcoin future expiring in December 2021. As of April 1st 2021, the price of bitcoin is 59095 USD. Let’s say the December 2021 future is priced at 62500 USD. The price of the future is higher than the price of the underlying. Let’s assume that when we go to 1st May, the bitcoin price is 61000 USD. The December 2021 future has now come to 62250 USD. Come 1st June, the bitcoin price is 61500 USD and the December future is 62125 USD.
In the above scenario, as the time moves, the price of the selected future gradually moves towards the underlying spot price. If the futures price starts higher than the underlying spot price and gradually decreases towards the spot price, it is called being in contango. If the futures price is currently below the spot price and it gradually increases towards the spot price it is called backwardation.
A futures curve either normal or inverted, is a plot of prices of multiple futures. A contango or backwardation is the movement of the price of the same future, over time.