When I was growing up, my grandparents used to use a word in Italian that sounds like "waglio". Whenever I used to see my grandmother, that is how she would greet me I'm not sure if I am spelling it right. Am I speeling it right? What does it mean in English?

it means "young man" and it comes from "guaglione" a napolitan dialect.

Guaglione:

YouTube - Aurelio Fierro "Guaglione" LIVE 1990
The song won the Naple festival in 1956

Can we get an English translation per favore?)

Staje sempe ccá, 'mpuntato ccá,
'mmiez'a 'sta via,
nun mange cchiù nun duorme cchiù
che pecundría!
Gué piccerí' che vène a dí
'sta gelusia?
Tu vuó' suffrí,
tu vuó' murí,
chi t''o ffá fá...

Curre 'mbraccio addu mammá,
nun fá 'o scemo piccerí'...
dille tutt''a veritá
ca mammá te pò capí...

E passe e spasse sott'a stu barcone,
ma tu si' guaglione...
Tu nun canusce 'e ffemmene,
si' ancora accussí giovane!
Tu si' guaglione!...
Che t'hê miso 'ncapa?
va' a ghiucá 'o pallone...
Che vònno dí sti llacreme?...
Vatté', nun mme fá ridere!

Curre 'mbraccio addu mammá,
nun fá 'o scemo piccerí'...
Dille tutta 'a veritá
ca mammá te pò capí...!

Nun 'e ppittá, nun 'allisciá
sti mustacciélle...
Nun cerca a te, nun só' pe' te
chill'uocchie belle...
Nun 'a penzá, va' a pazziá
cu 'e guagliunciélle...
Nun t'avvelí,
c'è tiempo oje ni',
pe' te 'nguajá!

Chi desidere 'e vasá,
scordatélla, piccerí',
ca si 'o ddiceno a papá,
chisá comme va a ferní...

E passe e spasse sott'a stu barcone,

Staje sempe ccá, 'mpuntato ccá,
'mmiez'a 'sta via...
Nun mange cchiù, nun duorme cchiù...
chi t''o ffá fá!...

You're always here rooted here,
in the middle of this street,
you don't eat anymore, don't sleep anymore
che pecundría!
hey kid, what are you saying
what's this jealousy?
you want to suffer,
you want to die,
who does oblige you...

Run into your mum's arms,
do not be silly, kid...
tell her all the truth
your mummy can understand...

And You keep on passing by this balcony
but you're only a kid...
you don't know women,
you're too young!
you're a kid!...
What do you have in mind?
go and play soccer...
What are these tears for? ...
Go away, don't make me laugh!

Run into your mum's arms,
do not be silly, kid...
tell her all the truth
your mummy can understand...

They are not for you [...]
these girls...
They don't look for you, they do not suit you
those pretty eyes...
Do not think on it, [...]
with the girls...
don't be discouraged,
there is time to come,
for you to get in trouble!

Who desires 'e vasá,
forget her, kid,
because if she tells her father,
who knows what will happen...

and You keep on passing by this balcony,

always stay here, rooted here,
in the middle of this street...
you don't eat anymore, don't sleep anymore...
who does oblige you...

This can be a rough translation, i hope someone from Naples can correct and complete it! :D

barcone (Neapolitan) = Balcony (Balcone in Italian)

"E passe e spasse sott'a stu barcone"
You keep (on) passing (by) under this balcony