You are talking about a branded "card" which comes from the knowledge graph.
In essence, you have to think in terms of RCS (real company [****] stuff) which is something Google says they looks for. Google will not take your word for it. In otherwords, what you do on your site is not enough.
At one point, I counted 46 branding signals. You do not need all 46, or even a majority, however, you do have to have enough of them for Google to feel confident that you are a real business with a brand. I will not list them all, however, I will give you a few of the major highlights.
Schema Mark-Up: You will need to properly mark-up your site using schema.org mark-up. You will need NAP (name, address, phone) at a minimum. This should be easily found just on click away on the About or Contact page. It can also appear in the sites header or footer.
Additional Contact Information: Often this is an e-mail address and/or contact form. This can also be additional names, addresses, and phone numbers with indications such as sales, marketing, etc. Again, this should be easily found one click away on the About or Contact page.
A Business Listing: Your company should be found on one of the respected business listing sites. While there are many, the trusted authority sites are ones that derive their data from other trusted sources primarily telephone listings from the local telephone company or one of the major sites such as yp.com which aggregates these listings. What is important to remember is that not all sites vet their data or derive them from valid sources. Stay away from these. They are not useful. Often, an ad in the yellow pages and ordinary business listing in the phone book will do the trick.
Google+ for Business: This is Google's opportunity to vet your company. By creating a Google+ business profile, you are giving Google a chance to ensure that you are who you say you are. Google will often, one way or another, validate any e-mail address, physical street address, and phone number.
Google Maps: Make sure that your business can be found on Google Maps. This requires a "walk-up" address and not a P.O. Box. Some have claimed that residential addresses no longer work. While an at home business is perfectly valid, Google may be using other metrics to ensure that a location of a business is legitimate.
Other Social Media: While I hate social media, it has become an unfortunate way of life. Make sure that you have some other form of social media account that can verify your NAP data. Facebook or LinkedIn are excellent for this.
Domain Name Registration: Google prefers that a business domain name registration be public with valid contact information. This is not a requirement, but is a significant factor. If your domain name is private, you can leave as it is. This is explained in the next section.
Consistent NAP Data: Google will check to make sure that all contact information found is reasonably consistent. This includes the NAP data found on the website, business listings, social media, registration information, etc. Enough consistent information will override a private domain name registration. If your contact information is not reasonably consistent, this will stop branding cold in its tracks.
Branded Links: It is one thing to have relevant link text and URL, however, some reasonable portion of your links should be branded with your business brand or name within the link text.
Branded Citations: Links are not enough. On other sites, articles, press releases, reviews, and other information should exist mentioning your brand or company name within the content and not within a header, footer, or sidebar.
Branded Co-occurrences: Citations are not enough. On other sites, articles, press releases, reviews and other information should mention your brand or company name as a comparison to another recognized brands within the content and not within a header, footer, or sidebar.
Branded Searches: Search users must search for your brand either as a Primary and Secondary search. A primary search is the first search entered by a user. A secondary search is where the primary search results are inadequate and a second more refined search is used. For example, athletic shoes and athletic shoes nike are an example of a primary and secondary search. However, athletic shoes and cheap shoes are not because these are two separate searches and not a refinement of the first search.
Wikipedia: While this is not necessary, an article in Wikipedia is certainly a significant factor. The editors will validate your business article to enough of an extent that Google feels comfortable with the listing. This also serves another purpose. Since Wikipedia is available via an ontology that Google relies upon on as a primary source of information within the knowledge graph, any article you create that is approved and published will immediately get picked up and used by Google. Wikipedia is a significant signal for branding.
These are the major factors. Do enough of this and you will get your card.