It really depends on the artist and the platform they're using. Some concerts include the livestream with the in-person ticket, but a lot of artists charge separately for it - typically you get a code or link after you pay for the online stream. There are several ways they do this: some use specialized platforms like Veeps or similar ones, others stream directly through YouTube or social media, and some even have their own subscriptions on their website. The thing is, there's no single standard - it really just depends on who's organizing it.
As for whether it stays up or not, that varies too. Some artists leave the video available for a few days after the concert, others keep it forever, and some only do a live broadcast and then delete it. Best thing to do is look carefully for the info on the event's website or ask directly before you pay - a lot of the time it's specified in the stream description. If it's through a platform like YouTube, there's usually a better chance it stays up, but not always.