The simplest way of connecting a camera to a microscope does not even need extra equipment. You only have to hold the camera at the right distance in front of the microscope’s eyepiece. You also have to make sure that the objective lens of your camera is smaller than the eyepiece of the microscope. Most digital compact cameras will work. Even the camera of mobile phones will work. Yo do need a steady hand, though. Zoom in and hold the camera a few mm in front of the microscope’s eye piece. The correct camera to eyepiece distance is quite important as is the correct centering and tilt of the camera. If you rest the camera against the eyepiece, then the distance is likely to be too short and you will not get an image. You then carefully have to move the camera away from the eyepiece, at the same time taking care that you do not tilt the camera, or move it sideways. THis requires patience. If camera and eyepiece are not properly aligned then you might get unevenly lit images, or no image at all. If you want to take many pictures, then I would recommend the use of a tripod. This method makes pictures in which you can see the whole field of view, much like looking through the eyepiece yourself. The picture is in a round circle, which is surrounded by a black border.
Having a camera permanently connected to the microscope makes life much easier. Here you have a third tube extending out of the microscope, which can be used to connect a camera. This will result in much more stable and reproducible pictures. Special USB camera can be directly inserted into the trinocular head and you can watch a live picture on the computer screen, but still have the eyepieces of the microscope available for direct viewing. Trinocular heads, however, are quite expensive, and especially educational microscopes do not have these.
Luckily there is a compromise. these USB cameras can also be inserted instead of an eyepiece. You first have to remove the eyepiece, of course. You can then watch a live image on the computer screen and take pictures by clicking the button of the computer mouse. The advantage of these cameras is that they no not require any specific adapters as they fit right into the microscope.
The better USB cameras can be quite expensive and often they still do not have the same high resolution as digital photo cameras of the same price. USB microscope cameras are simply not made to the same large volume as consumer photo cameras, and I think that this is one reason why the price is relatively high.
The most expensive method is to connect an SLR camera to the microscope over an adapter and intermediate optics. These are manufacturer specific solutions. The intermediate optics correct any lens errors and the image quality is one of the best which can be obtained. This solution is also one of the most expensive.
There are many microscopists who prefer to use single lens reflex (SLR) cameras over USB cameras. Many SLR cameras have another significant advantage: It is possible to make high quality videos with them! USB cameras are not able to transfer the data quickly enough through the USB cable and therefore the video quality is often much lower. If you use digital photo cameras, then there is also no need for you to turn on a computer. It is therefore not unusual for microscopists to make their own camera to microscope adapter, either out of solid cardboard or by asking someone for help.
SLR cameras also have a significantly larger sensor than USB cameras. The image quality is therefore better (better signal to noise ratio at low light intensities), and often the dynamic range of the camera is also higher. The camera has less problems capturing both bright and dark areas at the same time.
SLR cameras do have a disadvantage. Connecting them to the microscope requires a special adapter which are sometimes manufacturer dependent and expensive. They are also quite heavy and a trinocular microscope head is therefore necessary.