Traders create algorithms on the basis of some underlying hypothesis. Once created there needs to be a way to test out and validate the hypothesis.
Backtesting assesses the viability of a trading strategy by analysing how it would play out with historical data.
However, testing on historical data is not enough because the user might have overfit or underfit their trading strategy. As a result, before allocating capital to a strategy you might want to test it on live market data to get better results.
When a strategy is running in paper trading mode, it tries to simulate trading just like it would do in live trading except that no orders actually get placed. The strategy continues to compute results and shows the user performance on live data under the impression that all trades will get executed and thus giving users the ability to measure the performance of the strategy without making any investment.
You can learn more about paper trading here.