Dimethylethylamine or Ethyldimethylamine?
There's a technicality in that the 2013 IUPAC recommendations (Nomenclature of Organic Chemistry: IUPAC Recommendations and Preferred Names 2013 (Blue Book)) have slightly awkward preferred IUPAC names for simple amines like these. The preferred IUPAC name for your molecule is "N,N-dimethylethanamine" (as described by P-62.2.2.1 on p 671, but note that there is an erratum for the triethylamine example shown there).

Putting that aside, though, what you're conceptually missing is that the methyls and the ethyls in your proposed name are (implicitly) not on equal footing. The methyls are names of substituents, but ethyl is not a substituent - it is the parent chain (since it is longer than methyl). So, $\ce{CH3CH2NH2}$ is named "ethylamine", and $\ce{CH3CH2N(CH3)2}$ is named as a N,N-disubstituted form of "ethylamine", i.e. "N,N-dimethylethylamine". In common usage the "N,N" is often dropped, leaving "dimethylethylamine". I should emphasise that this is not in line with the IUPAC recommendations, but people can and will use it, and this is why you will see it on the Internet. Returning to the preferred IUPAC name in the previous paragraph, we can see that the use of "ethanamine" rather than "ethylamine" makes the difference between methyl and ethyl a lot clearer.

In fact, according to the recommendations, it is also permissible to name tertiary amines in the manner (alkyl1)(alkyl2)(alkyl3)amine (although it does not lead to the preferred name). If we were to adopt this method, then the name would be ethyl(dimethyl)amine, because now there is no notion of a parent chain and all three substituents are merely substituents - which should be listed in alphabetical order ignoring prefixes like "di". The parentheses around "dimethyl" are required (but not around "ethyl") because of a separate rule (P-62.2.2.2, p 672 - see also the discussion in the comments).

To sum up:


N,N-dimethylethanamine: Preferred IUPAC name
ethyl(dimethyl)amine: "Alternative" IUPAC name; permissible but not preferred
dimethylethylamine: Doesn't comply with 2013 IUPAC recommendations, but the logic behind the name is similar to that used in generating the preferred IUPAC name.