First of all, let me say that none of these photos are awesome or amazing or anything spectactular. I took my 8 year old outside for literally 5 minutes to get these shots specifically for this blog post.
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There are many things that make a good photo, but I'm just going to address a few of the basics.
If you are looking for a photographer, make sure they can at least get the basics correctly!
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FOCUS!
There is nothing worse than capturing a great moment with your camera, only to find out later that it's out of focus. Yes, it even happens to professionals from time to time (which is part of the reason we take so many photos in the first place).
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This photo is properly focused. The eyes are perfectly in focus.

This photo, while viewed small, looks like it is fairly in focus, but if you click on it and then click again you'll see that the eyes are completely out of focus and that the focus fell on the hair.

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EXPOSURE!
This is a little tougher for some people to recognize, but it is definitely one of the most basic things in photography.
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This photo is properly exposed.
There are a couple of bright spots on his shirt, but his skin is properly exposed.

This photo is overexposed, meaning it is too bright.

This photo is underexposed, meaning it is too dark.

FINDING THE LIGHT!
I always say that the most important thing in photography is light. Part of being a photographer is knowing how to find locations with good lighting.
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This photo has good lighting.

This photo, taken only INCHES from the one above, has dappled light. The spots on his skin and hair are caused by the light shining through the trees above. This is not good lighting.

This photo is backlit, meaning the light (the sky in this case) is behind him. While backlighting can be done properly and with beautiful results (using a reflector, fill flash, etc.) this is NOT an example of how to do proper backlighting.

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Another thing you will notice about all of the photos above is that they have an out of focus background. This is a result of the depth of field that I created using my camera settings. Good portraits generally do not have the subject (person) and background both in focus.
The subject needs to pop out of the background.
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There are, of course, many more things that go in to a good photo and even more still that go in to making an amazing photo, but these are some of the basics that I thought I would address today.
You can also apply some of these basics to your own photography when taking your own family photos!