The explosion actually occurred in New Jersey. The Black Tom explosion was set off by German saboteurs during WWI. Black Tom Island was an artificial island (made of fill) connected by a causeway (with a rail line) to Jersey City and approximately 2/3 of a km away from Liberty Island (since then much more of the area between Black Tom Island and Jersey City has been filled in and become land). During WWI the Island had served as a sort of ad hoc munitions depot, where train cars and barges filled with explosives and ammunition were kept in preparation for being shipped overseas to WWI combatants, as the US was still officially a neutral party in 1916.

Due to the naval blockade by the allied powers US munitions had become much less valuable to Germany while still being incredibly useful for the allies. This provided motivation to strike at munitions facilities in the US, which the Germans did on several occasions during the war. The Black Tom explosion involved about 900 tonnes of explosives and ammunition being set off. Details of the explosion are still somewhat unknown to this day though there's a strong consensus that German saboteurs were responsible.

As one might imagine a nearly 1 kiloton explosion is quite a "sporting" event, and it properly devastated the Black Tom terminal and caused broken windows and structural damage up to several kilometers away. The Statue of Liberty was quite nearby, however, so it sustained a fair amount of damage, including some structural damage to the torch arm, which is why that part has been closed off to tourists since then. Fortunately, fires that had been set on the pier had caused most of the depot to be evacuated so only 4 people were killed in the explosion, as far as we know, though hundreds were injured.

It's questionable to call this a terrorist attack per se, though many terrorist attacks did occur around the turn of the 20th century, in the US and in Europe.