Change font size of label with data defined expression in QGIS?

Firstly, go on the Layer Properties > Labels > Text window.

Then, edit the expression for the text size (I marked it in the image below):

In the expression field, simply write the name of the field which stores the sizes for the fonts of the labels. In my case, the field was called font_height, so I wrote (remember to use these " "):

Finally, click on the Apply button and this will be the result:

In @mgri 's answer "font_height" will need to give a valid value for the size field (i.e. a number or number-like string). If it is a number you can simply use "font_height" as the expression. If not you will need to do something with the value. If "font_height" had values of 'big', 'medium' and 'small' you would need to use some conditional logic:

CASE
    WHEN "font_height" = 'big' THEN 14
    WHEN "font_height" = 'medium' THEN 10
    WHEN "font_height" = 'small' THEN 8
END

Alternatively, if "font_height" values were something like '14pt' you would need to something like:

replace("font_height", 'pt', '')

Just to add to mgri's answer, you can use CASE WHEN at the same location:

CASE WHEN [column-A] = [something] THEN 3
     WHEN [column-A] = [something else] THEN 4
     ELSE 5
END