Google+ has a lot going for it as a place to share and show photos. It has nowhere near the active user base of Facebook, but it has more photographer-friendly features.
When using images of your Google+ profile page, the image sizes are generally more logical than the mishmash of image sizes that is a Facebook Page. But there are still some sizes to know and tricks to getting the images to look their best.
Google+ Cover Photo
The cover photo is the large one at the top. It's displayed at an aspect ratio that's essentially 16:9 (or very close to it, at least), the same as the HD video standard. On desktop screen sizes, your image is displayed at right. At left is an automatically semi-transparent overlay under your profile picture.
Because Google+ uses a responsive layout that adjusts for different-sized screens, the actual size that it's displayed at varies quite a bit. The largest your image displays at is 1080 x 608.
The semi-transparent part at left also resizes, and it duplicates part of your image.
The ideal size for the image you upload is 1080 x 608 (aspect ratio of 135:76, which is very close to 16:9). The smallest image you can use is 480 x 270 (aspect ratio of 16:9). The largest image you can use is 2120 by 1192.
As you reduce the width of the browser window, the cover photo scales accordingly, until at a certain point when you make the browser very narrow (as it is on a smartphone), the layout switches to a different mobile-friendly layout, like this:
JPEG Image Quality Issues with the Cover Photo
Like the other social media sites like Facebook and Twitter, Google+ applies some pretty heavy JPEG compression to the cover photo. That reduces the image quality, can make text blurry, and tends to make colors less vibrant.
But there's a solution. Rather than upload a JPG, upload a PNG-24 image. Those aren't converted to JPEG. They get some PNG optimization applied, but it will stay as PNG.
Be aware, though, that a PNG-24 creates a much larger file than a heavily compressed JPG image. So it's going to be that much slower for your users' browser to open the page. It's also possible that Google could close this little loophole at any point.
Google+ Profile Picture
Google Plus uses a round profile picture. On your profile page, it's shown as a circle with a 120px diameter.
While nothing is stopping you from using any image for your profile picture (as long as it doesn't violate Google's terms of service), Google recommends that you use "a good, recognizable headshot."
Google+ Profile Picture and Google Search Results
Google was, for a time, showing thumbnails in search results if you set up what is known as authorship links. But that little experiment has ended, and Google is no longer supporting authorship markup.
When a link thumbnail displays at full-width, it displays at 426 x 255. The jagged edge overlay at the bottom is applied automatically--there's no way to remove it.
But not every link you post will include a full-width thumbnails.
There's a list of requirements that the post you're linking to has to meet before that happens. Criteria include that the page has photos meeting minimum width and height requirements (506px and 303px), a maximum ratio, as well as that the page (or site) is set up to include various tags in its code. You can find further details here.
If those requirements aren't met and yet there's still an image on the page that can be used for a thumbnail, it's fit within a box 120px by 120px, like so:
If there's no image in the post, the link box will just contain text.
It's not clear to me how Google chooses the initial image to suggest--it's not necessarily the featured image if you're using WordPress, and it doesn't seem to necessarily be the images tagged in the Open Graph code. When posting and editing you will often--but not always--get the option to scroll through some thumbnail choices when you post the link. To access that, roll over the thumbnail image in the upload dialog. If arrows appear over the top left of the image, use them to change the thumbnail.
The layout that Google+ uses is responsive, which means it will adjust to the viewer's screen size. On wide displays, like a 27-inch desktop display, it will put up to three columns across, like this:
On something like a 15-inch laptop display, it might be only two columns across, like this:
And on a narrow display, like a smartphone or even just a narrowed browser window, it'll go to a single column and the cover photo layout switches, like this: