How to name this organic molecule?

The IUPAC name is methyl pentanoate.

The trivial name is methyl valerate.

this is a carboxylic acid functional group
.. ... . O
.. .. ...||
.. R - C - O - H

this is a ester
.. ... . O
.. .. ...||
.. R - C - O - R'

esters are named similarly to carboxylic acids.  start with the acid and replace the ending with -oate. (or -ate in some cases).  

.. ... . O
.. .. ...||
.. H - C - O - H..... formic acid

.. ... . O
.. .. ...||
.. H - C - O - CH3..... methyl formate.... (the CH3 group is the methyl group)

.. ... . .. .O
.. .. ..... ..||
.. CH3 - C - O - H..... acetic acid

.. ... . .. .O
.. .. ..... ..||
.. CH3 - C - O - CH3..... methyl acetate

.. ... . .. ... ... ... ... ... .O
.. .. ..... .... .. ... ... .... .||
.. CH3 - CH2 - CH2 - C - O - CH3..... methyl butanoate (note the 4 carbon chain)

butane is CH3-CH2-CH2-CH3.. 4 carbons

pentane is 5 carbons in a row.... that's what you have.. so what would you call it?

IUPAC says "methyl valerate".
Other possibilities are "methyl ester of pentanoic acid" or "methyl pentanoate"