You may already have a Matter controller. For WiFi Matter devices you may be able to use your smartphone.
For iPhones (and iPads) just make sure you have OS version 18. They actually added Matter controller functionality in 16.1 but many people found it very flakey. Before 18 when I tried it I got my smart plug (Tapo TP15) connected and it worked fine for something like a day and then disappeared, and then I was never able to get it connected again. Even after telling the phone to forget the device and doing a factory reset on the plug I could not get the phone to see it. 18 on other hand has been fine.
For Android phones you need at least Android 8.1 and Google Play services 22.48.14 or later and Bluetooth Low Energy (BLE) 4.2 or later. That's what the net tells me--I do not have any direct experience with Android. The net is also telling me that you really want Android 12 because Matter support was not as stable in earlier versions.
From the name Matter controller and the diagrams in many explanations of how Matter works people tend to get the impression that a Matter controller is kind of like the WiFi router on your WiFi network: some dedicated device which you generally only have one of.
To see why that is not the case it is helpful to look at what a Matter controller actually does. (The general idea is right in what follows, but I'm sure I've botched some details that don't affect the overall points).
When you start out with a brand new Matter device and power it up the device will come up ready to be paired with its first controller. The controller and the device communicate using BLE, but the device will only talk to a controller that has a code that is hardwired into the device. That's what is in the QR code that is usually printed on the device or its packaging or a slip of paper in the package.
The controller and the device use that code to establish a secure communications channel over BLE. The controller gives the device WiFi connection information and gives it a certificate that can be used to verify that future communication over WiFi is with that controller. The device can actually remember several of these controller certificates, allow the device to work with several controllers.
Later, when things want to communicate with the device over WiFi it will only talk to things that can prove they have the private keys for a controller certificate the device has.
There is nothing that requires those private keys to only be in one place. If you set up a new Matter device using your iPad, for example, it will store the keys in Keychain and they will be shared with your other Apple devices, such as your iPhone and your Mac. Your iPhone and Mac can then authenticate to the device and it will accept commands from them. As far as the device is concerned they are all the controller that set it up.
There are some advantages of getting a controller that is not your phone/tablet/computer, such as a HomePod or Apple TV. Those support remote control through the cloud. If you just use your phone/tablet/computer you can only control the device locally.
Suppose that you set up the device first using an iPhone as the controller. Later you want to also be able to control it from an Android phone. What you do is go to Apple Home, find the device and open its settings, and somewhere in there there will be an option to put the device in pairing mode.
Select that and Home will generate a one-time pairing code, give that to the device, and put the device in pairing mode. It will also tell you that code.
You then go to your Android phone and go through its process for setting up a Matter device, except you use the code you got from Apple Home instead of the code that came with the device. The code that comes with the device is only used to set it up with the first controller (initially or after a factory reset).
That gets the certificate for your Android phone onto the device. It doesn't disturb the certificate from the iPhone so now the device will recognize both your Apple stuff and your Android phone as legit controllers.
Now suppose later you get an Amazon Echo and want to add the device. Echos can act as Matter controllers for the Alexa ecosystem. You can use any one of the existing controllers that are paired with the device (iPhone or Android phone, or anything those have shared their private key with like iPad or Mac) and have that put the device in pairing mode again and generate a one-time code, and then use the Alexa app on one of your phones to go through Amazon's Matter controller setup.
In summary:
1. Each ecosystem (Apple Home, Google Home, Amazon Alexa, Home Assistant, Samsung SmartThings, etc) that you want to use a Matter device with needs a Matter controller.
2. A Matter controller is not necessarily a separate piece of hardware. If you just want to control a Matter device from your phone you've probably already got everything you need.
For Matter devices that use Thread for wireless communication instead of WiFi, you are more likely to need hardware other than just phones and tablets. I think some newer iPhones do support Thread directly, but mine doesn't and I don't own any Thread devices so have not looked deeper.