I have a magento website and a wordpress website.
I have tried using the fishpig extension to integrate the two, however the functionality is limited and I lose my theme and design when using it.
Is it possible to display the wordpress site in the magento webstore?
Essentially retain the Magento header and Footer and in the content area just put the wordpress site?
First of all, be aware that when you have Magento and Wordpress on the same server, you're opening an additional vulnerability leak. When your Wordpress is being hacked, the hacker might also be able to get into your Magento installation and database. That way, the hacker can collect your customers' personal information and maybe even creditcard information.
But about the question you're asking: When using Fishpig's extension, you will lose some functionality, but you can do the theming of the blog within Magento, so it will look the same as the rest of your shop or completely different, depends on what you want.
Another option might be to get rid of Wordpress and use an extension like AW_Blog instead. That way, you'll move your blogs into your Magento installation. Downside is, you have to migrate all blogposts and you'll lose more features that aren't available in the extension.
Last but not least, you might want to consider running Magento and Wordpress as two separate platforms and use Single Sign-on. This means that you login once, no matter if that's on Wordpress or Magento, and you're logged in to both platforms. I'm not a Wordpress expert, but I know there are several Magento extensions available to do this.
Arjen is entirely correct in saying that an install of wordpress and magento on the same server make you at risk of hackers gaining access to both sites. If you are diligent about updates and making sure you have fully vetted all your wordpress extensions, you can safely do this. But just know there are some risks.
The most seamless integration i have used is this wordpress extension:
Do note that this does require a core override on the magento side. But that being done, you will be able to pull in your header and footer into the wordpress side. It will maintain your customer's cart, session and cookies. It's not the easiest integration to pull off, but it works great once you have it running.
The other solution i have used is to use the wordpress api to get blog data from the wordpress side to the magento site. In this case the wordpress blog had to be on another server. I had to rebuild the magento header and footer in the wordpress theme, and then use the api to have the blog titles populate in the magento header with the latest posts. I also has to get the post excerpt, featured image and title on the homepage.
This one required a lot more work, but you end up with two sites that are safe from one another, and no customer data is at risk.