Offshore means not in your country. It could be as simple as wanting servers in the same country as your customers. Or it could be for tax-evasion purposes. Or it could be to do or sell something illegal in your country and make it look like it was coming from a country where it's legal. Like selling marijuana seeds to the USA from a website hosted in Amsterdam.
Onshore server You can host anything that is legal for example adult content.But not warez such as mp3 Offshore server You can host anything that prohibited by onshore hosting such as warez.But no to child porn.
"Off the shore (of your country)", works literally for countries that are also islands, but it's extended to the others as well. Therefore: "abroad". However, as far as hosting, it's often used to mean "a service in countries with different laws, that basically allow or don't care about what isn't allowed in another country". Therefore: "in more accomodating countries"; ie: Sweden, Holland, China, but all countries could qualify depending on each customer.