So let's say that you are a very rich man and you want to hire a driver for your limousine. I think that everyone being in this position would check thoroughly the background of a candidate for this delicate and critical for his safety position. I am sure that the rich man would like to know when the candidate got his driver license or if the candidate has ever been caught driving drunk, how many road traffic code violations have done etc. The rich man asks the candidate for those pieces of information during the interview and the candidate responds that he has everything in a text or spreadsheet file in his computer and he is more than willing to share it with him. The big question is: Would the rich man believe the candidate or not ? The candidate claims that everything in this file is true but only he and the authorities can know for sure if it is true or if something is missing because it is created by himself. The solution for the rich man would be to contact the authorities (a third party) and learn the truth. There is no other solution which does not involve third parties that can guarantee that something (a block of data for example) is true and/or it has not been tampered by someone.
And that's where blockchain comes in.
Blockchain in layman's term is a distributed database that guarantees with a cryptographic way the continuity and the immutability of the records written without the need of a trusted third party. Data is organized in timestamped blocks and each block has a unique hash which has been calculated for the specific data that this block holds. Each new block added in the blockchain checks the validity of the hash of the previous block and thus the data in a block cannot be tampered or deleted simply because the hash will change and as a result the new block will not be added to the chain. A public blockchain is available to everyone and thus everyone can see the content of its blocks.
So in the example above if each and every fact regarding the history of the candidate driver was written in the blockchain by the responsible party (authorities such as the Ministry of transportation, police etc), the rich man would be sure that it is true and it has not been tampered and he would actually have a mathematic proof of that because we know that it is computationally impossible to calculate the same hash for a different block of data. The rich man would be able to see the blockchain himself and by whom each fact in the driver's history has been written by. So, the rich man would be sure for the truth!