An inroad is an attack or raid.
To make inroads is to move toward something or advance on a goal at the cost of something else. Basically, one can make inroads to enemy territory. Reading the examples will be especially helpful on this phrase. In the singular form, this definition of the noun is commonly used with the verb give (e.g., gave him an inroad). While, as noted above, in the plural form it is commonly used with the verb make.
In short, the difference between the two is largely the difference between figurative and literal raids or attacks. It should be noted that the plural form is much more common.
Examples
In particular, the desire for a voice became the inroad for Islamist movements into tribal areas, which created a new type of tribal activist and parliamentary representative independent from tribal chiefs loyal to the government. [The Epoch Times]
Longtime suitor Bee Taechaubol had an offer accepted for €470m to purchase 48%, thus giving the Thai businessman an inroad to begin investing in the side after clearing the club and holding company Fininvest’s financial checks. [Football Italia]
The Islamic State, also known as ISIS, is making inroads from the east, where it seized the desert city of Palmyra last month. [The New York Times]
Netflix has steadily made inroads into the movie business, promising to upend its traditional theatrical window system and more quickly debut movies to its 60 million-plus subscribers, following a limited release in theaters. [Press Herald]