The Button doesn't recognise any text. The Button doesn't even know the TextBox exists. In order to do something when the Button is clicked, you need to handle its Click event handler. To do that, you will generally double-click the Button in the designer. When you do that, the code window opens and you see the generated Click event handler. That event handler is a method (Subs and Functions are all methods) and it's a member of the form, not the Button. When the user clicks the Button it raises its Click event, which invokes that method of the form. That's the sum total of what the Button does. Everything else is up to the form.
So, in order to get this all to work, the first thing you need to do is set up your ImageList correctly by providing a key for each Image. If you are adding the Images in the designer then the Name property of each one is its key. If you're adding them in code then you need to pass a key when calling Add, e.g.
ImageList1.Images.Add("Chicken", Image.FromFile("Chicken.jpg"))
Now, to display a random Image you should create a Random object and generate a random number and use that as an index into the Images collection of the ImageList. You get the Image at that index and display it. Just make sure that you keep the index, which you would do by assigning it to a field.
When the user clicks the second Button, you call the IndexOfKey method of the Images collection of your Imagelist and pass the value they entered, i.e. the Text of the TextBox. If the index of the key matches the index of the Image you displayed then the user has correctly identified the Image, otherwise they have not. Note that IndexOfKey is case-insensitive so you don't have to worry about upper and lower case in your keys or the user's input.