All the other answers work fine. I too suggest you to use Illustrator when you can, especially Rishab.Ag's tips in his answer; it's really the fast way to do it!
But maybe for some reason you need to do it in Photoshop and I want to add a ultra-simple trick that you'll probably use in many other situations.
PS: This is not a solution if your logo is an object, but a good solution if you really copy/paste your logo in Photoshop and
simply want a quick way to fix it.
From my understanding, you want to get rid of the white ONLY in the graphic, not the text.
1 - You can use the Magic Wand Tool, with a tolerance of 64 (see top menu for this). Select a white area anywhere in your image.
2 - Then, go in the menu "select", and choose "similar". All whites will
3 - After this, click on the Rectangular Marquee Tool
While holding the "option/alt" key, trace a rectangle around the white text in your logo. When you do this, it tells Photoshop to remove what you're now selecting from your total selection.
This means you will only have the white from the graphic selected after this.
The green rectangle shows what to select while holding the "option" key down. Even if you forget some areas or didn't trace a perfect rectangle, just trace another one while holding the "alt/option" key; it will remove anything you'll select until you stop pressing that key!
4 - Finally, use the "Eyedropper" tool, select the black color of your background, and press the "X" key to make it switch to the background color in your swatches; then simply press the "delete" (backspace) and "command" at the same time. The selected area will be filled with the background color you selected.
Pressing "X" is just a shortcut to make the foreground/background colors switch position. And pressing "delete/backspace" + "Command" is another shortcut to fill the selected area with the background colors from the swatches. I guess it's like a paint bucket (I never use paint bucket and such...)
If you prefer to remove the white color, simply click "delete". You can always add a black layer under this one too to keep that transparency for later.
Additionally, on smaller logos, sometimes the wand tool does a very bad job at selecting the pixels and anti-aliasing. You can either increase your "tolerance" to include more pixels OR you can always expand your selection by 1 pixel to grab more anti-aliasing.
Is the result above what you were looking for? This is one way to do it. Using the alt/option key to remove some selection is very useful! I actually prefer to use the technique above instead of clicking every single area... It's way faster and you can use the lasso tool or polygonal lasso tool as well to deselect some areas.