Google's search results are personalized even if you don't log in to a Google account. Google uses your location and your previous searches to customize the results. When you log in to your account, Google has more information: your search/browsing history, your social connections, the pages you've bookmarked or +1'd. Everyone is different and Google has the opportunity to use information about you to disambiguate queries, to show more results from the sites you trust and pages you're already visited.
Google+ can change Google results more than any other Google service because social data is an important signal for personalizing results, Google owns the end-to-end experience and it's in Google's best interest to show that Google+ helps sites get better rankings. After all, the reason why many sites added Google+ buttons is that Google+ data is used by Google's search algorithms.
The latest initiative that integrates Google+ with Google Search is called "Search, plus Your World." It's an upgrade of the social search feature that integrates the rich content from Google+.
Google Search now has access to non-public posts and photos from Google+ and you'll be able to search all the posts and photos shared with you or from your circles. "You can find relevant Google+ posts from friends talking about an amazing trip they just took, whether they've shared privately with you or publicly. You'll find links shared by your friends, such as activities, restaurants and other things they enjoyed on their trip," mentions Google. Personal results have a special icon next to the snippet, Google shows the number of personal results at the top of the page and you can also restrict the results to these pages.
Google Instant suggestions now include the people from your Google+ circles. "Now, typing just the first few letters of your friend's name brings up a personalized profile prediction in autocomplete. Selecting a predicted profile takes you to a results page for your friend, which includes information from their Google+ profile and relevant web results that may be related to them." Google shows similar suggestions for "prominent people from Google+, such as high-quality authors."
Another change is that Google includes a special box with Google+ pages related to your query. "Starting today, if you search for a topic like [music] or [baseball], you might see prominent people who frequently discuss this topic on Google+ appearing on the right-hand side of the results page. You can connect with them on Google+, strike up meaningful conversations and discover entire communities in a way that simply wasn't possible before," informs Google. I think it's a mistake to show Google+ pages that are vaguely related to a general query like [music]. The box looks more like a thinly disguised ad than a genuinely useful feature (an ad both for Google+ and the celebrity pages).
To balance the increasing number of social features, Google also added two new buttons at the top of the search results pages that help you toggle between the personalized results and the non-personalized results. This works for an entire session and you can also remove personal results from the preferences page.
Right now, "Search, plus Your World" is gradually rolled out over the next few days, but it only be available when you use Google.com in English and you sign in.
While Google+ data can certainly improve search results, I feel like there's too much information that's not really useful and too much Google+ bias. When Google developed the OneBoxes for maps and stocks, it linked to its own services, but also to competing services. Today Google no longer tries to be fair. Showing the number of people that added the author of a news articles to their circles is not more useful than showing the PageRank of the page or the number of Twitter followers. Showing the latest Google+ posts below the homepage of a business is not more useful than showing the latest Twitter posts. Google profiles are not necessarily better than Facebook profiles and the number of +1's is not more useful than the number of likes. To make Google+ more powerful, to attract more celebrities and businesses, Google might end up making Google results less useful. It's a tricky balancing act to use Google search's popularity to increase Google+ adoption, while also improving Google results using Google+ data and there are many mistakes to be made along the way.