The emoji in question are cartoon-like graphics depicting men holding hands, women holding hands, and same sex couples as families. Those emoji are available along side similar graphics of mixed sex couples.
Apple's emoji collection caters to the openminded and tolerant crowd
You can see the mixed sex family emoji in one column, followed by same sex with women, then same sex with men. Another column includes mixed sex and same sex couples holding hands, along with two women wearing bunny ears holding hands and dancing. In the bottom right corner there's a smiling pile of poo, which apparently doesn't bother Russia's government.
The law that spurred the investigation was enacted in 2013 to target anything the Russian government deems “nontraditional sexual relations.” The emoji were added to the iPhone, iPad, and iPod touch as part of iOS 8.3 last April.
The police investigation started in August, according to the Russian news site Gazeta (english translation). Assuming Apple is found guilty of giving pro-homosexual materials to minors, the company could face fines between 800,000 and 1 million rubles (between US$12,200 and $15,250), and a three-month sales suspension in the country.
Russia is notoriously anti-LGBT, and has drawn international criticism for its hardline stance. That position hasn't, however, stopped Russia's political leaders from using Apple products despite the inclusion of the emoji their police are now investigating—or because Apple CEO Tim Cook is openly gay.
Despite Russian officials willingness to use Apple products, some did push to ban iPhones and Macs from the country because they don't support the idea of a company run by an openly gay man. So far, nothing has come of that and Apple continues to sell its products in the country without any problem.
Considering Russia's attitude towards the LGBT community, maybe emoji to show up in the iOS 9.1 beta could serve as a political statement from Apple.