Well, I think 'to enjoy' is pretty good translation of it. A milder version of it would be 'to like' (pitää) something, although there's a difference that enjoyment is often something related to an act in progress. (I like coffee vs. I enjoy coffee (right now)).
You could say it's more intense because the verb "nauttia" is often used to describe things like sex, culinary treats and other inanimate things that one may <em>really</em> enjoy, and not just "like". In that sense the word "nautinto" can often be translated as pleasure and not enjoyment.
The word is less frequently used when talking about people, although you could say "Nautin seurastasi" ("I enjoy your company").
I'll give a few examples of usage for the word so you can think about it further. Some are sort of secondary uses of the word, but should exemplify that the word has some different connotations.
<em>Minä nautin aamukahvini parvekkeella. = I enjoy my morning coffee on the balcony.</em>
<em>Hän nauttii syytesuojasta. = He enjoys immunity from prosecution. </em>
<em>Minä nautin kesästä. = I enjoy the summer.</em>
<em>Minä nautin joka hetkestä. = I enjoy every moment.</em>
<em>Seksuaalinen nautinto. = Sexual pleasure.</em>
<em>Nautintoaine = stimulant (e.g. coffee, drugs, cigarettes, alcohol...)</em>
<em>Nautin kahvia = I enjoy coffee (right now, as in drinking it)</em>
<em>Nautin kahvista = I enjoy coffee (in general)</em>
<em>Pidän kahvista = I like coffee (in general)</em>
<em>Pidän kahvia (kädessäni) = I'm holding coffee (in my hand)</em>