I want to overpower the sun, but due to my budget, I couldn't get a monolight. I have to make do with the Nissin Di700A I bought. When the flash is near the subject and faces the subject straight on, it can overpower the sun — but I want to get that nice soft lighting without losing too much power. Could I achieve that with a softbox or would I lose too much power?
Yes, it is possible.
Two factors need to be considered: The power of the flash and the sync speed of your camera.
Flash power (Guide number) is being discussed in other answers. Consider that adding a softbox reduces the effective power of any flash. That is because the softbox spreads the same amount of light over a greater area, but also because part of the light is absorbed by the materials of the softbox. This means that if you measured the "guide number" of the flash+softbox combo, it would be a significantly lower number than for the flash alone. By how much? I depends on softbox size, design and quality of materials.
Balancing continuous and pulsed light
Remember that correct exposure is the result of proper use of 3 variables: aperture, shutter speed and sensitivity (ISO).
Aperture and sensitivity affect both continuous light (the sun) and pulsed light (the flash, strobes). However, shutter speed affects mostly continuous light, as usually a flash pulse is way shorter than the time the shutter remains opened.
That means that if you choose a short enough exposure time, the contribution of the continuous or ambient light to the final exposure is low enough as to allow the flash to dominate the scene. The lower your flash power, the faster you need the shutter. This is one way of explaining the technique to balance continuous light versus flash exposure.
The limiting factor here is the maximum speed at which the camera con both, fully open the shutter and fire the flash. This is known as "Sync Speed".
You may find the maximum speed either in your camera's user manual or spec sheet or, set the camera to fire the flash (i.e. not auto flash) and try to increase shutter speed. Chances are the camera won't go as high as usual.
Mechanical shutter cameras tend to have a lower sync speed, while other types of shutter can go very high. (For example my Canon 50D has a Sync Speed of 1/250 sec, while my Olympus XZ-1 can go all the way up to 1/2000 sec, it's overall maximum shutter speed).
If your gear does not let you set the correct values, consider using shades, diffusers and light bouncers instead. There are factory made "Photo reflector disc" that you can buy (rent or borrow) in several sizes, but they can also be made with home or office available materials.
Set your scene differently or at different time of the day. Early in the morning and before sunset ("The golden Hour"), the sunlight is less powerful but also falls at an angle, which can be used at your favor in many ways.
Use light bouncers to fill-in the shadows and reduce contrast.
If shooting at mid day, put a diffuser above the subject, and a bouncer or reflector on the side to fill-in and create volume.
Revise this post with corrected GN for 100 ISO setting. GN for 200mm lens = 175 GN for 105mm lens = 160 GN for 35mm lens = 90
Using data for a 105mm lens setting:
The guide number for this unit is 160 ISO setting is 100. We use a guide number by dividing the distance, flash to subject into the distance in feet. Suppose the camera is set to 100 ISO and the subject distance is 12 feet. The aperture setting is calculated as 160 ÷ 12 = 15. We set the exposure aperture to f/16.
Suppose the ISO setting is 200, the guide number now = 320. For this set-up the aperture for a subject at 12 feet is 320÷ 12 = 26, we set the aperture to f/22.
OK, now that you know how to use a guide number, know we can also work them backwards.
Say the camera is set to 100 ISO and the sunlight exposure reading is f/16 @ 1/125 of a second. To match the sunlight intensity, we divide the guide number by the f-number. Thus 160 ÷ 16 = 10. This tells us to place the flash 10 feet from the subject and the flash intensity will match the sunlight intensity.
Let me add that in my opinion, you should set the flash 1 f-stop subordinate to the sunlight. This will establish a 3:1 lighting ratio which is deemed best for most portrait situations. To accomplish, we multiply the syncro-sunlight distance by 1.4. This re-calculates the flash to subject distance making the flash subordinate by 1 f-stop. Thus 10 x 1.4 = 14. Setting the flash to subject distance at 14 feet does this deed.
Conversely, if you want the flash to overpower the sun by 1 f-stop, divide by 1.4. Thus 10 ÷ 1.4 = 7. Set the flash to subject distance to 7 feet will overpower the sunlight by 1 f-stop. .