Note: I have already tried googling. Although similar questions have been asked on different forums, I couldn't find a detailed explanation, which I could really understand.
Circuit diagram courtesy of Wikipedia.
This is very easy to understand why centeraltap transformer is needed in a full wave rectifier. Let us assume that we have a simple transformer, and there are two diodes and the central wire coming out from the transformer is not present there which is obvious since we are not using centeraltap transformer. So now see the figure
In first case let A be at lower potential i.e. negative and B be at higher potential i.e. positive. So due to induction 1 will become positive and 2 will become negative. Now since 1 is at higher potential so D1 will be forward biased and current will flow through it but since 2 is at lower potential D2 will be reversed biased and no current will flow through it so the circuit will not work because in one case D2 will be reverse biased and in other D1 will be reverse biased. Now if we take a centeraltap trnsformer as shown in figure
there will be following sign convention in case 1 and just opposite in case 2. So in case 1 as you can see D1 is forward biased so current will flow through D1 and R and will go to the central low potential point and the circuit will be completed for case 1 and in case 2 the current will flow through D2 and R and again will go to the centeral low potential point and the circuit will again be completed and it will rectifier the full wave that is it's negative as well as positive half cycle in a positive variable DC current. So that is the reason a center tap transformer is used in a full wave rectifier because you need to rectify full wave. Hope it helped you.