Lately, I've seen lot of bloggers looking to integrate their Instagram photos in a nicely formatted way within their blogs and websites. Generally, 
SnapWidget - This is one of the most popular Instagram widgets used by large number of users. It can be included within any website with ease.The free version is quite flexible with loads of customization options. You can either pull photos from an Instagram account or can use different hashtags to fetch relevant images.
The widget itself can be displayed in different ways viz., slideshow, grid and many more. Since this widget is responsive, it can nicely fit in almost every type of layout. One can also apply various special effects to customize the look and feel of the widget as well as to make it interactive. Once configured, you can easily get the widget code that can be easily pasted into your website layout.
Intagme - This is yet another instagram widget that's not only easy-to-use but is also feature packed. Quite similar to other popular solutions, this one also lets the user fetch photo streams either from an account or through hashtags.
Thankfully, you can also specify the size of the individual photos within the grid. This way, widescreen layouts can also use this Instagram widget quite easily.
Instansive - This responsive instagram widget is one of my favorites. It has a nicely formatted grid photo showcase that is ideal for embedding in the sidebar.You can also specify the number of columns and rows to be included in the photo grid. One can also include social sharing buttons within the widget so that your Instagram feed can be easily shared across different platforms. You can also show photo captions if required.
WP Instagram Widget - As the name implies, this is a flexible plugin for showcasing your Instagram photo stream. You can also customize the look and feel of this widget through custom CSS code. It also uses caching to avoid any latency while the widget is loaded within the page. There's no authentication required to fetch the photo stream making things easy to install and configure this handy widget. If you're looking for a simple Instagram plugin for WordPress, then this widget is ideal for you.
InstaEmbedder - This is perhaps the most flexible instagram widget with good number of options to customize the widget at will. Users can mix both hashtags and account photo streams together to get a richer grid of images. You can also fine tune the grid photo spacing, photo size, and the borders. You can even customize the font and caption colors making it match closely with your website theme.
Imageagram - The hover effect supported by this instagram widget is quite impressive. Same goes for image padding that can create different types of photo grids.Since photo colors, as well as the widget background, can be easily changed through the settings, you easily transform its look and feel to match with your website design.
Instafeed - This instagram solution is ideal for power users who have working knowledge of JavaScript. It's a pure JavaScript solution without any dependency on an external library. Simply instantiate a slideshow object with mandatory options and place it at the desired place. The available options are good enough to bend the photo grid at will which can also be customized through custom CSS and HTML.
Websta - This instagram widget can be easily integrated within XHTML templates like blogger blogs. The widget creation process is simple and includes several customization options.Apart from accounts and hashtags, you can pull popular Instagram photos to create a regularly updated photo grid. You can get up to 150 pictures in a single gird with custom spaces and borders. Specifying a background color for the entire widget ensures it gels well with the theme of your website.
Pic Puller - If you're using the popular ExpressionEngine CMS for powering your website, this flexible instagram add-on is an ideal option for you.It comes in two flavors and the free version is good enough for most users with ample customization options. The integration and setup of this add-on are very easy and you can showcase your account's photo streams in different ways on your website.