Understanding Public Charging
People like simplicity and standardization, and while to those of us who have owned electric cars for years we understand where we can (and cannot) charge, to new members of the Electric Vehicle community it can be VERY confusing.
Overall - this post will split between Tesla vehicles and non-Tesla vehicles, and is focused on charging in the North America.
1) Generally any Tesla vehicle can charge at any Tesla Supercharger, or any Tesla "destination charger" - I say "generally", as there are very specific exceptions like some salvage vehicles that cannot charge at a Supercharger, and the original Tesla Roadsters are a completely separate ball of wax which won't be covered here
2) In the U.S., generally Tesla's CANNOT charge at non-Tesla fast chargers (also called DC chargers). These chargers specifically use a "CCS" or a "CHADEMO" adapter. There are a few exceptions to this:
Tesla in the past sold a "CHADEMO" adapter - if you have one of these, you can use a fast charger that uses CHADEMO - note, before you rush out to buy one of these (they were fairly expensive - $400), they charge at a max rate of 50 kWh (or somewhere between 150-200 miles of range added per hour) - far slower than most Tesla Superchargers
There are a few "CCS" adapters for Tesla's to be found on the Internet. At this time, I have not seen one offered by a well known company, or sold through a well known e-tailer. Also, the reviews indicate a max charge rate of 50 kWh, and these adapters are pricey. Tesla sells a CCS adapter in some other countries, but there are different CCS standards worldwide, so these adapters don't work in the U.S. There is reason to think Tesla will offer a CCS adapter in the U.S. at some point in the future
3) In the U.S., Tesla's can charge at non-Tesla destination (or home) chargers. These chargers use a J1772 connector, and all Tesla's are sold with a J1772 adapter (so you need to be sure you have the adapter with you). In the past there were relatively few non-Tesla fast chargers, and many "destination" chargers, so many Tesla owners could take for granted, that they could charge "anywhere" - but non-Tesla high speed chargers have proliferated in recent years, making the assumption no longer correct that any charger can charge a Tesla
How do you know if a non-Tesla charger is high speed vs. destination? Technically, the high speed chargers use either CHADEMO or CCS connectors, and the destination uses J1772 connector. Another tell tale sign is if the charger says "DC" its a high speed charger. Finally, apps that locate chargers like PlugShare can filter by your car/type of charger. Of course, the last way is that the "CHADEMO" and "CCS" connectors simply will not fit in the Tesla or any of the adapters that come with the Tesla!
1) In the U.S. right now, non-Tesla vehicles CANNOT charge are Tesla Superchargers. Some people will purchase an adapter for the Tesla charger connector and think this will work with Tesla Superchargers - it will not. Tesla's use the same connector for fast charging (Supercharger) as for destination charging, so this can be confusing for non-Tesla owners. Tesla has announced that it plans to allow non-Tesla's to charge at Superchargers in the future - this will likely require an adapter, and will likely only be for avaialble for non-Teslas that support CCS (most of the latest non-Tesla's support CCS. The Nissan Leaf is an exception)
2) non-Tesla vehicles can charge at any non-Tesla destination charger. All of these destination chargers use J1772, and all non-Tesla electric vehicles support J1772
3) non-Tesla vehicles with a CCS port can charge at CCS high speed chargers, and non-Tesla vehicles with a CHADEMO port can charge at CHADEMO high speed chargers. This may sound confusing, but its not as complex as it would seem, as most non-Tesla high speed chargers support both CCS and CHADEMO, and really the future in the U.S. for non-Tesla's is CCS (so unless you have a Leaf, you likely have a CCS port). To know if a charger support CHADEMO or CCS, you can look on the charger (it will say), or use a charger locator app like PlugShare
4) non-Tesla vehicles can use Tesla destination (or home) chargers, if they have bought a Tesla to J1772 adapter. Be sure when you buy an adapter, you buy the right one - the J1772 to Tesla adapter is designed for Tesla vehicles, as it takes a J1772 input, and converts it to Tesla. The Tesla to J1772 takes a Tesla port input, and converts it to J1772 for the non-Tesla car.
How do you know if a Tesla charger is a Supercharger or a destination charger - Superchargers are typically very distinct looking - see the picture below. The Tesla destination chargers will look like they could easily be in a person's garage.
Hopefully I have not thoroughly confused you! As noted above, there are many apps for finding chargers. I use PlugShare, and one nice feature with PlugShare is you can enter your vehicle and model, and filter on the available chargers based on your vehicle.