This is a screen capture of the garden blog A Succulent Life and as you can see the image that Julie added to personalize her blog's header doesn't fit within the constraints of the header. If you have this problem on your own blog there are two things you can do to fix this.
Step #1: Save or Test
Whenever you're going to perform any modifications for your blog the first thing you should do is backup and save your template. To achieve this click on Layout>Edit HTML and click "Download Full Template." If anything should happen you can always upload your template and restore your blog. Another option I like is to create a test blog where you can perform these modifications and get your feet wet and comfortable with the process before you do it to your garden blog. You can never be too careful.
Step #2: Locate header wrapper
A Succulent Life is using the standard Minima template without any modifications that I can see. This is what the header wrapper currently looks like:
margin:0 auto 10px;
border:1px solid #cccccc;
The reason that the image doesn't fit within the borders of the header on A Succulent Life is because the image is too wide for the 660 pixels that have been assigned for the header. We have two options available to us to fix something like this.
Re-size the image so that it fits within the constraints of the header. When uploading an image to your header make sure that it isn't as wide or wider than the width that has been allotted to it. In this case the image shouldn't be wider than 660 pixels.
If you don't want to go through the trouble of resizing the image again then just add enough pixels to the header wrapper so that your image fits comfortably. Change the width (660 pixels) to whatever the width of your image is and then Save your template.