As others have mentioned, it depends on the specific brand and model.
Wi-Fi doesn’t have a fixed set of standard commands the way Zwave and zigbee do , so it’s not just a matter of being on the same network. SmartThings would have to know the exact message content that that particular device expects.
Typically you can get integration in one of five ways now and a future 6th, depending on the brand and model:
the future is coming: Matter
A) The home automation industry is very well aware of the frustration that consumers have trying to figure out what will work with what, and they are working on a new and very important inter-operability standard called Matter. All of the big companies are on board, including Amazon, Apple, Google, IKEA, Philips Hue, Samsung, Aqara, and a whole bunch more. (The notable exception being Tuya). Seriously, if this happens the way it’s supposed to, eventually the answer is going to be super simple: buy a model with the matter logo and it should work with SmartThings. 
And some of these are gonna be cheap generic Chinese devices. You’ll just need to look for the matter logo.
Consumer reports has a very good FAQ on Matter
But there are only a few of these available now. Leviton has one, Tapo has one, but there aren’t many. And you do have to have a smartthings/aeotec hub to use any matter device with SmartThings, so it’s not a solution for everyone.
FAQ: What do I need to add a Matter device to the SmartThings app? Do I need a bridge router device?
But I need something before Matter devices are available! Like, now!
OK, then you need to look into the following possibilities.
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the device manufacturer publishes an “open API,” which is documentation describing the exact messages that their device expects. These are the manufacturers that want to encourage integrations, such as Phillips hue with their bridge. If the manufacturer has an open API, it will usually be described on their website, again because they are trying to encourage integrations.
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The device manufacturer has an official integration with SmartThings. There are some of these, like iHome, Currant, Meross, TPLink Kasa, and Globe Electric. In that case, it will be listed by brand in the SmartThings app.
- The device has an IFTTT service/channel, and then you can get simple integration that way. For example, miLight has one.
Smartthings will never see these directly.
Instead, you can create a “virtual“ switch to act as a proxy for the real switch.
Then you use Ifttt with four “applets“:
- when WiFi switch turns on, turn on proxy switch
- when WiFi switch turns off, turn off proxy switch
- when proxy switch turns on, turn on wifi switch
- when proxy switch turns off, turn off WiFi switch
OK, that gives you a way to look at your smartthings device list and see whether the Wi-Fi switch is on or off. It also lets you turn it on or off from smartthings, because when smartthings turns the proxy switch on, Ifttt Will turn the Wi-Fi switch on.
This is the basic method for use with any Wi-Fi on/off device that has an ifttt channel.
And, yes, it’s a lot of work to set up because you need to create those four applets for every individual Wi-Fi device you want to control.
And it really only works well for on/off. It’s not very good for dimmers or for color changing. There may be some ways to make that work as well with a similar method, but it depends on the exact details of the device and it can be a lot of work.
If the Wi-Fi device has scenes that show up in its Ifttt channel, you might also be able to use those. But again, that varies a lot from model to model.
So that’s the basic Ifttt approach. Create a simple on/off virtual switch to stand in as a proxy for the Wi-Fi device. Then create four different Ifttt applets to keep your proxy device and the real Wi-Fi device in sync. 
Note that while Ifttt used to be free, as of September 2020 you will have to pay a monthly fee if you have more than three applets which means using this method will require a paid subscription with them.
- The device manufacturer has an integration with the Smart Life or Tuyasmart apps.
In that case, you can probably get some integration through those apps. See the community FAQ:
FAQ: How does TuyaSmart Integration work? [Updated 2023]
- Using Alexa as a “man in the middle”– works better for pocket sockets than for Wall switches.
If the device works with Alexa routines (not SmartThings routines), you can set up a virtual sensor in SmartThings and then use that to trigger the Alexa routine. The problem is if someone turned the switch off at the wall, smartthings won’t know it. ( that may change in the future as Amazon is adding new features all the time, but for right now, it’s a one-way integration)
- The device works with HomeKit. If so, If you’re already running HomeKit and you’re willing to run an additional computer as a “man in the middle” server, you can set up a Homekit simulation, called homebridge, and get some integration that way. This is the more technically difficult Then the first four methods, but there are other community members who will be able to help you if you are interested.
OK, those are the fairly straightforward ways.
Other Options
Then, as has been mentioned, if none of those five are true it sometimes happens that a community member has figured out what messages the device is expecting and has set up some kind of integration. These often require a strong technical background to set up, and may require an additional “man in the middle” server like a laptop or a raspberry pi. ( or in the case of Sonoff – based devices, flashing the firmware to change it to something that is more compatible with SmartThings.)
To find those, usually the easiest way is to go to the quick browse lists in the community – created wiki and look on the “pocket socket” list in the device type handler section. If an individual thread is tagged " requires server" then it’s one of the ones that is going to need an additional server device and some programming skills.
You can also check the same quick browse list to see if there’s a simple integration that’s already been done. Or just search this forum directly.
FAQ: Using the quick browse lists without the community wiki
So it just comes down to the specific brand and model. 
If you already have an MQTT broker running, you can try the following
Or if you are willing to set up home assistant, that can sometimes be used as an intermediary, but home assistant itself requires additional devices and considerable technical skill.