Resolution is about how much detail you can see. It determines how far apart two things must be in order to be resolved as separate images.
Objectives and condensers have ratings for their Numerical Aperture, or NA. This tells you how good their resolution is. The higher the number, the better the resolution.
For dry objectives and dry condensers, where no immersion oil is used, and only air comes between the slide and the lens, the maximum NA is 1.0, and the maximum NA that is actually realizable is closer to 0.95.
The resolution, measured in nanometers of separation between the two objects we are trying to resolve, is calculated by the following fairly simple formula:
The wavelength of the light is generally white light, centered at 500 nanometers (green). The resolution is thus given in nanometers.
Suppose both the objective and the condenser are in air, and the NA of the objective is 0.65, and the resolution of the condenser is also 0.65. The formula then becomes:
Doing the math gets us 516 nanometers of resolution, which is about the best you can get with a 40x objective with an NA of 0.65. If two things are closer than half a micron, they will look like a single object.
The condenser's NA should match the objective's NA. This is because the condenser is projecting a cone of light onto the objective. If the condenser's NA is too low, the spot of light will not fill the entire width of the objective lens, and the objective will not have its full NA. If the condenser's NA is too high, light from outside the field of view will enter the objective, and reduce the contrast. You control the NA of the condenser by using the condenser iris to reduce the circle of light to the right size.
Unless your condenser has immersion oil between it and the slide, its maximum NA will be 0.95. And if your objective is an oil immersion objective, but you are using the condenser dry, then the condenser will not be able to fill the view of the objective, thus limiting its NA to 0.95 also. So the maximum resolution will be:
which comes out to be 353 nanometers of separation.
If your oil immersion condenser has an NA of 1.30, and your objective has an NA of 1.30, then you can get a resolution of 258 nanometers.
These calculations reflect the theoretical best resolution, and should be thought of as upper limits, rather than what you can actually achieve.
Using a shorter wavelength of light will get us better resolution. We can use an LED or the laser from a BluRay disk player, with a wavelength of 405 nanometers, and get an upper limit resolution of 190 nanometers using the NA 1.30 optics.
In the photo below, we have four yeast cells, outrageously over-magnified in order to make the resolution comparisons that follow more easily seen. The photo was taken using the 10x objective on one of my microscopes.
There seems to be some amount of detail barely discernable inside the cells, but we only get a hint of it.
Moving up to the 20x objective, a little more detail is seen, but not a lot more. Something appears to be darker in the center of the cells, but that is about the limit of what we can say.
Using the 40x objective, we start to see more than a single dark spot inside the cells. The upper cell seems to have two dark spots, and maybe a third beneath them, but it is hard to say.
Switching to the 100x oil immersion objective, and adding a little immersion oil to both the condenser below the slide and the objective above the slide, we get to the limit of this microscope's resolution. We've been cheating a little bit in this series of photos by using a blue filter under the condenser to make the light more monochromatic, to eliminate some of the chromatic aberration. So our calculations of the resolution assume a shorter light wavelength than the normal 550 (green) that is used for a white light.
We can now see four dark spots inside the upper cell, and the space between the four cells at the center is resolved into something approaching a square instead of a circle.