Most of the time Spanish follows Subject - Verb - Object order or what you will commonly see abbreviated as SVO order. In certain situations, however, Spanish reverses that order. To keep this answer brief, I am going to restrict it to just those instances where subject and verb are reversed and omit discussion of objects since that aspect of word order doesn't apply to the example you have given us.
In general, Spanish doesn't usually end a sentence of this nature (and by that I mean a sentence that includes a restrictive clause) with a verb. Your sentence —
A la gente mayor le encanta ver los tatuajes que lleva la gente joven hoy en día.
Older people love to see the tattoos that young people wear today.
provides us with an example of such a sentence.
A la gente mayor le encanta ver los tatuajes ...
is the main clause and:
... que lleva la gente joven hoy en día.
is the restrictive relative clause. It cannot stand on its own as a sentence. It is dependent upon and descriptive of some element in the main clause (in this case "tatuajes"). In such a clause, the part that follows "que" often uses Verb - Subject order. I believe you can find several sources that make mention of this, but finding one that was easy to use for teaching purposes was a bit difficult. I thought the best of what is easily available and free was found on Wikipedia. Here is an excerpt from its page on "Spanish grammar," specifically, its section "Order of constituents":
In many dependent clauses, the verb is placed before the subject (and
thus often VSO or VOS) to avoid placing the verb in final position:
Este es el libro que escribió mi amigo, but rarely Este es el libro
que mi amigo escribió = "This is the book that my friend wrote"
One source lists several reasons for reversing SVO order. I'll list some of them for you here. These come from the book, Spanish Grammar in Context, 2nd edition:
Spanish is much more flexible than English with regard to word order
in sentences. Although the verb usually follows the subject, as it
does in English, for emphasis or focus, different elements within the
sentence can be placed in initial position. This includes the verb
which can precede the subject.
Empieza la caza del meteorito.
The hunt for the meteorite begins.
[The sentence above] could be rewritten [placing the subject first]:
La caza del meteorito empieza.
but the emphasis achieved by having the verb in initial position is
somewhat lost, at least in writing.
With short sentences, the tendency is to place the verb before the
Murió su madre.
His mother died.
If the subject is much longer than the verb, again the tendency is to
have the verb in initial position.
Respondieron Carmen y su marido, y un par de personas más.
Carmen and her husband, and a couple of other people replied.
Within a clause, the verb is normally put before the subject.
... lo que pretenden los científicos ingleses ...
... what English scientists intend to do ...
I didn't do an exhaustive search to see if a question like this has been asked before, but I did come across this:
Why is the order of the object and the action reversed in some cases?
It isn't exactly like yours and the answers are a bit different, so perhaps you will find it useful as well.