Mount Vesuvius is perhaps best known for its eruption in Roman times (24 August 79 AD), described  by Pliny the younger, when Pompeii and Herculaneum were destroyed. The eruption left a large crater that can still be visited today. It is a currently dormant volcano that occasionally emits streams of lava (last minor eruption in the 1950's; although its dormant state could be an indication of a build-up of pressure and a coming explosive eruption). At the top of the volcano is a crater rim that affords a view into the crater that still fumes slightly. Besides that you will have (only on clear days, but the mountain is notoriously covered in fog or clouds) a stunning panorama overseeing the Bay of Naples, Naples, Capri, Ischia, the edge of the Amalfi Coast and more.
Be warned: Going there will make you sweat a lot and set you back about 15€ (including private taxi up the hill).
By Public Transport
From Naples: There's a direct bus-line that leaves twice a day from Naples Central Station square and goes straight up to the visitor parking. It's run by one of the minor bus companies (not the city transport) and you might have a bit of a hard time finding it, but unless you have a pre-arranged group tour it's perfect. It runs on a timetable, so it's less capricious than the taxi service from Ercolano. Two-way ticket from Naples Station runs at € 13.50. Bus leaves at 09:20 and 10:35 at the Garibaldi station.
From Ercolano: Take the Circumvesuviana railway (Sorrento-Naples line). From Naples buy tickets on the lower level of the Central Station aka Stazione Centrale a Piazza Garibaldi, or bus to Ercolano Scavi station in Herculaneum or the Pompei Scavi station. From there you have two choices:
You can either wait for the local bus that costs around €8.70 return (from Pompeii, cheaper from Herculaneum). The bus goes only around twice a day from Herculaneum - at 8:45 AM and 12:45 PM. It goes more often from Pompeii.
The tourist information office is closed on sundays.
You can also do a 6km hike to the summit along the winding roads. To do this you will need to take the local bus service ANM 5 or 176 from Ercolano to San Vitto. From there you will be able to walk to the summit. You can take in the views of the national park and enjoy the local wildlife. You will still need to pay the entrance fee when you arrive at the crater, but for the energetic ones it is a must!
You can get to the upper parking area, but be prepared to part with at least €2.50 for the parking fee. And it's a good idea to go there earlier in the morning to avoid the jams that may happen along the narrow road.
Note: there are a couple of huts selling cheap and sometimes bizzarely inappropriate souvenirs and expensive drinks around the rim. Take a bottle of water with you to avoid these excessive charges.
Weather: even on hot days you can get strong and chilly winds at the rim so take a lightweight fleece to wear at the top.
Eat & Drink
Vesuvius is an active volcano and may erupt any time in the near future. Over the last few centuries, Vesuvius has erupted at intervals ranging from 18 months to 7½ years, making the current lull the longest in 500 years. Its lack of eruption may merely indicate a build-up of pressure, which may result in a more explosive next eruption, posing a lethal hazard to over 500,000 residents living in the same place that got destroyed in 79 A.D.
Listen to local authorities if there is any news about the state of Vesuvius, and do not wander any further than the trails go.