Macro to find cell value in a column

[wwballar42]

I don't have much knowledge with VBA so I appreciate all help I can get.
I'm trying to write a macro to do the following:
1) Take cell A1 and copy adjacent cell B1 in new sheet.
2) Then search cell B1 to find a match in column A.
3) If there is a match, I want to copy that adjacent cell in B and into the new sheet.
4) If there is no match, then leave as is from before and move onto the cell.
5) Continue for the rest of the data.

Here is an example of what i am trying to achieve
This is in the first sheet:
Column A Column B
Car Saw
Dew Jacket
Pen Key
Saw Screen
Hand Shoe
Jacket Window
Screen Sock
Saw Ride


Now I want this in the next sheet:
Column A Column B Column C Column D
Car Saw Screen Sock
Car Saw Ride
Dew Jacket Window
Pen Key
Saw Screen Sock
Saw Ride
Hand Shoe
Jacket Window
Screen Sock

I hope this helps and thank you for your help!

I will attach what I am looking for.