Learn Finnish: Onko tuo kuppi lasinen? - Is that cup made of glass?

It comes from the noun lasi (glass) plus the suffix -nen. In Finnish, adding -nen to a noun is a highly productive way to create an adjective that means 'made of [material]'. You can do this with almost any material, like puu (wood) becoming puinen (wooden).

I learned earlier that materials take the partitive case. Could I say Onko tuo kuppi lasia instead?

Yes, absolutely! Both Onko tuo kuppi lasinen? and Onko tuo kuppi lasia? are perfectly correct and mean exactly the same thing. Finnish freely lets you choose between using a material adjective ending in -nen or using the partitive case to describe what an object is made of.

How would the adjective lasinen change if I wanted to say 'from that glass cup'?

Adjectives must match their noun in case. For words ending in -nen, the basic form drops the -nen and switches to a stem ending in -se- before adding case endings. So, to say 'from a glass cup' using the elative case (-sta), lasinen becomes lasisesta: tuosta lasisesta kupista.

Could I just use a compound word like lasikuppi?