Star Wars: The Force Awakens History
Twenty years after the Imperial defeat at Endor, the New Republic has emerged as the leading government in the galaxy, but the remnants of the Galactic Empire still remain in the form of the First Order. The Republic has sufficient military strength, mainly in its navy, to defend against direct attack by the First Order, but it apparently lacks the political will to attack and eliminate its rival. The Republic does, however, support the Resistance, which serves as a kind of proxy to combat the First Order.
While not seen in the film, the governing institutions of the New Republic are presumably similar to those of the Old Republic, suggesting a senate with representatives from member worlds with a Chancellor chosen from among them acting as head of government. It's reasonable to assume that checks have been added to prevent the kind of takeover that Palpatine managed.
The number of planets controlled by the New Republic was not specified, but it is presumably more territory than the First Order controls. The state of the Republic in the aftermath of the destruction of its capital (presumably Coruscant) is unknown.
At the time of signing, only around 40 planets openly declared their support of the reformed Republic. Despite initial claims, it would take them three years to take Coruscant and establish themselves as the leading galactic power. Even then the Republic had only 600,000 member worlds by it's height, small compared to the 12 million member worlds currently held by the Empire. It would take them less than 25 years after that to re-establish the Old Republic's ineffectual power structure, just in time for war with the Yuuzhan Vong. This demonstrated just how pathetic the New Republic had become and would result in the formation of a more federal system of government, which eventually became a new Galactic Empire.