For a social media platform provided internationally, do I have to apply for patent protection in multiple countries?

Patents are territorial, meaning that patents only provide protection in the particular country in which they are granted. So if you obtained a US patent, you only have protection in the US - a US patent will not provide you with any protection outside of the US. There are effectively "exceptions", such as, if you file a European Patent Application, you can obtain a patent which is effectively valid within a group of countries, however, you have to at least file a patent with the European Patent Office. So, again, patents are only valid within a country, or a group of countries, in which you have actually obtained the patent.

Two great answers already. I just want to add that a patent in the US (for example) would give the patent owner the right to exclude others from making, using, offering for sale, selling, or importing into the United States the invention claimed in the patent. Therefore, if a competitor copies your patented technology, they would be infringing by simply marketing or offering it for sale in the US.

As Steven mentioned, it is entirely based on where you want your technology to be protected. You will need to file in every area that you want protected. Initially, you can file a PCT application that will grant protection in up to 148 countries. However, for long-term protection, you will need to eventually file for specific country/region protection (national filings) no later than 30 months after the earliest filing date of your initial application.