Why does my camera hesitate when shooting?

What you are describing is shutter lag. When you press the shutter release, the camera must focus before exposing the image. There are many ways of avoiding this. The easiest is to half-press the shutter so that the camera focuses, then press fully down when you want to take the image. Because the half-press will focus, there is less or no lag when you fully press the shutter release.

Another method is using back-button focus. Here you set up the camera to focus using a button on the back of the camera, and the shutter release is set up to take an image whether or not the subject is in focus.

If your subject is moving quickly, then you could switch the focus to AF-C mode. This will track the subject as it moves and you should have little lag when you press the shutter. Again you'll probably need to half-press the shutter for the autofocus system to start tracking. This is for a D5500, but may apply to the D3300: How do I select AF-C on a Nikon D5500?

It takes a little time between the moment you decide to press the shutter button and the picture is actually taken. Part of this is due to your "reaction time" and part of it is due to the camera's "shutter lag."

In order to capture a precise moment, you need to learn to anticipate the moment you wish to capture and start the process that leads to the shot being taken just enough in advance of that moment so that the shutter is actually open when the precise moment occurs. You also need to plan ahead as much as possible and do everything that can be done ahead of deciding exactly when to take the shot.

There are several things you can do to reduce your reaction time and the camera's shutter lag:

• Watch your subject through the viewfinder. If you wait until whatever it is you want to take a photo of is already happening before you raise the camera to your eye, you'll probably be too late.
• Meter the scene and adjust the camera's general settings for things such as ISO, aperture, metering mode, etc. in advance.
• If you are using a zoom lens, zoom in or out to the focal length you think you'll use to actually take the shot, rather than zooming along with the subject as it approaches or moves away from your position (unless you are specifically going for 'zoom blur').
• Activate metering and autofocus before you decide upon the precise moment you wish to take a photo. This is typically done by 'half pressing' the shutter button. When the electronic information that shows things like shutter speed and aperture light up in the viewfinder, that is an indication that metering is active. Depending on how you've selected certain options in your camera's menu, autofocus is usually initiated along with metering by a half-press of the shutter button. For advanced techniques used to separate metering and AF from each other and/or from the shutter button half-press, please see the links at the end of this answer.
• If the light is changing constantly and you are using an auto or semi-automatic exposure mode, choose one that continuously updates exposure until the image is taken. Some exposure modes "lock in" the exposure settings that are set when metering is first activated and the shutter button is continuously half-pressed.
• Select an autofocusing mode that tracks your subject if they are moving. For Nikon and a few other brands this is called AF-C (AF-Continuous), for Canon it is called AI-Servo AF (Auto Intelligent Servo AF).
• Continue to hold the shutter button half-press until just before the exact moment you wish to capture. Press it down the split-second before the desired "moment" that the camera needs to open the shutter.

There's an old saying that has been around action and sports photography circles for a long time: "If you saw it happen, you didn't get (capture) it." This is because with SLRs and DSLRs the viewfinder is "blind" while the image is actually taken. There are a lot of sports/action shooters that shoot with both eyes open in order to see:

• The wider area around the camera's limited angle of view (it can help to keep from getting clobbered on the sideline as well as help with situational awareness regarding photographically interesting things going on in the game).

• What happens during the split second their viewfinder is "blacked out" while the image is captured.

So aim to press the shutter button just early enough that you don't see, via the viewfinder, the exact instant in time you want to capture. With a little practice, you will get a feel for just how long that is with your particular camera and the way you have it set up.

Advanced camera models in particular can have so many user selected options that the timing can be significantly affected by which options are or are not selected. With entry level cameras such as your D3300, there are fewer options that change the amount of time the camera needs to do what you tell it to when the shutter button is pressed all of the way. But most entry level cameras are typically slower than advanced cameras, so that forces the user to anticipate the precise moment they wish to capture just a tad bit sooner than the user of a faster handing and more advanced camera.

For further reading:

What does the AE/AF lock button do that half-pressing the shutter doesn't?
How can I enable back button focus and disable focusing with shutter button on a Nikon D5500?
Why isn't my DSLR focusing accurately on a fast-moving subject?
Why my "action" shots are blurry even shooting on AF-C, is this a lens or camera limitation?

Try switching the lens to manual focus.
If the picture is taken directly then there is two possibilities.

1. you have a slow focusing lens, some are really slow and can be a pain to use.
   Generally going with a normal brand sigma, Nikon, Tamron etc. You have a fast or medium fast lens.
   It's only when you buy a really cheap lens/brand you can expect anything else.
2. You do not focus prior to taking the picture.
   This is done by half pressing the shutter button on a standard configurated camera or with the button you have assigned focus to.
   You can also set the AF mode to servo.
   This will continually refocus as you half press the shutter button (or other assigned focus button), this is good for moving objects.

The common situation would be if you are using flash and have Red Eye Correction turned on. Auto mode will use flash indoors. Red Eye Correction waits one second while it flashes early, trying to make the subjects eye pupils contract, but this makes the shutter one second late.

Turn Red Eye Correction off, and the picture will be immediate again.

See your camera manual Index for Red Eye Correction.

Other possibilities for delay would be if you have the Self Timer enabled. This will be a little longer, 2 seconds minimum.

Make sure you have set exposure delay to off. It might be the reason why it is taking time to shoot a picture. This happened with me too and took me some time to realize why it took too long.

As others have suggested the main culprit can you be the shutter lag because of the time taken to focus your subject.

You can reduce the lag by clicking ahead of your expected shot. Eg. If you are expecting to take picture of a person in air, press the shutter when the person starts to jump.