Copy Names from Column A to Column B If Those Names Are Found In A List Of Names

It would be pretty simple to run it off IF(COUNTIF>0) except that you're looking for words in side a text string, and it's not an exact match because the "DATABASE" range has an extra space on the end and the "NAME" range don't.

I was looking at maybe doing an arrayed formula like {SUM(SEARCH)>0} to replace the COUNTIF, but then I ran into the problem that you would find the string "Ada" inside the string "Adalbert" and that would be that.

I think you'd be better off doing some pre-processing, if possible:
1) Excise the training spaces from the DATABASE column by running CTRL+H and replacing all the empty spaces " " with a null string ""; all you have to do is put a blank space in the FIND box and leave the REPLACE box empty.

2) Use the "Text To Columns" command (Data ribbon; leftmost button in the Data Tools group) and split out the NAME fields in A:A so that you've got first names in column A, second names in column B, third names in column C, etc.

Then you can run something like IF(COUNTIF(database, name1) + COUNTIF(database, name2) + COUNTIF(database, name3) > 0) to do your test.