Cut out logo from white background

jozo,

From the screenshot I believe you have created it as vector artwork, as you should, in which case you can just Export or Legacy Save for Web as PNG24 with Transparency.

The following may sound unforgivably outdated and boring, sorry.

If you wish to have PNGs (PNG24, of course) look crisp and clean, it is important to have the images in the exact desired final pixel x pixel size, forget about resolution which may actually lead to wrong sizes and hence blurriness. It is easiest and safest to work at the desired size.

A very common unsuitable way is to Export to PNG (remember to use PNG24 and use Transparency for artwork to be in front of different backgrounds) with a medium or high resolution, such as 300PPI.

To retain the clean and crisp artwork, a PNG must be created at exactly the pixel x pixel size that it is to be used for (or second best at sizes that are powers of 2 times as large).

For clean and crisp artwork avoid JPEG.

It is also important to have the artwork and also the Artboard placed fully within integer/whole pixel X and Y values in the Workspace. Otherwise the resulting image will become a bit wider/taller and the extension(s) will be empty and therefore be white.

Therefore, the safest way is to create the artwork at the final pixel x pixel size and use a corresponding Artboard, then either use the Legacity Save for Web (where you can look in the Image Size window for size confirmation) or Export at 72PPI. In either case, use the relevant optimization (available with both ways); it is also convenient to have 72PPI in the Effect>Document Raster Effect Settings.

If you have pure vector artwork, you can relax a bit and have the artwork/Artboard at any size (the Artboard must have the same proportions as the final image), then use the Legacity Save for Web and set either Width or Height in the Image Size and Apply (make sure the other value is also correct).

The Legacy Save for Web may be an old carthorse, but it knows its way home, even if the driver is drunk and sleeping it off in the hay in the back.

There's several ways to do it but I have a video here that might help you out:

https://youtu.be/XCO-W_gbiqA

Pay close attention to 2:30 where you can choos a trasnparent background.

If this doesn't work for you, please share a screen shot with your logo selected and the Properties panel displayed. 

Is the logo a placed raster image? Based on the fact that you're saying it has a white background, I'm guessing that it is.

Please take a screenshot of your layers panel. Does it look like the one below?

If so, and it's a simpl logo, you can run Image Trace on it. Go to Window > Image Trace. And choose Color. 

Then click Expand in the Control Bar.

Then when you view in Outline Mode (View > Outline), it should look like below. Switch to your white arrow and grab around the edge of the outside bounding box Press delete twice.

Then go back to View > Preview. Ungroup, and then you should have access to all the different bits and pieces. Now, if your logo is a low-quality image, or has a ton of colors, of compiund paths (like in the center of the pencil below), it might not work perfectly, but this should get you going in the right diretion. If you need more help, please post screenshots of your logo and your layers panel so we can better help you.