The Eastern Road (also known as the High Road) is a major highway in the Seven Kingdoms. It begins at the Crossroads Inn where the highway intersects with the Kingsroad. From the crossroads in the northern Riverlands it continues on into the Vale of Arryn.
Loosely speaking the Eastern Road is essentially a continuation of the River Road, and together they form a major highway for east-west travel across the continent, from the Vale in the east all the way to Lannisport in the Westerlands, passing Riverrun in between. However, the name of the highway changes when it passes the Kingsroad so that the two halves are considered two separate roads, probably due to their drastically different topography (wide and wet river paths versus treacherous and narrow mountain paths).
From the Crossroads Inn, the Eastern Road only runs a short distance east through the northern Riverlands before it enters the Vale of Arryn. The road leads from the crossroads to the only major pass through the Mountains of the Moon that form the Vale's border. The road is very dangerous in the mountain crossing, weaving its way through rocky narrow turns, and travelers are often assaulted by barbarous Hill tribes and prowling Shadowcats.
Eventually the mountain pass comes to a narrow choke point which forms the only way through the mountains, guarded by a strategically vital fortress known as the Bloody Gate. After passing through this fortress the road enters the Vale proper, and travelers are much better protected from dangers. The commanding position of the Bloody Gate allows House Arryn to effectively shut down the Eastern Road and block all overland travel into the Vale at their whim, which also presents a significant obstacle to invading armies.
Soon after passing through the Bloody Gate, the road arrives at The Eyrie, the ruling seat of House Arryn. The road then continues on to Gulltown, the main port of the Vale and fourth largest city in Westeros.
When Catelyn Stark takes Tyrion Lannister prisoner at the Crossroads Inn, she loudly and frequently announces in public that she intends to take him back to Winterfell via the Kingsroad. However, after traveling for some time she has the hood over Tyrion's head removed, and he recognizes that they are actually traveling along the Eastern Road to the Vale and the Eyrie. Catelyn falsely announced that they were heading back up the Kingsroad to Winterfell to throw any Lannister forces searching for Tyrion off their trail. On their way, they are attacked by some of the hill tribes, and they lose several members of their party.
Later, after Tyrion is freed at the Eyrie as a result of a Trial by combat, he returns back west along the Eastern Road with Bronn the sellsword in tow as his bodyguard. While on their way back along the road the two are set upon by other hill tribes led by Shagga, Timett, and Chella. Using a mix of charm and intelligence, Tyrion manages to convince these hill tribes to side with House Lannister and march to the Riverlands to join his father Tywin's army. Tyrion pointed out to the hill tribes that they are the enemies of House Arryn but can't afford good quality weapons to mount a serious attempt at conquering the Vale, but the wealthy Lannisters can provide them with quality weapons and siege engines. Moreover, as the current ruler of the Vale is the sister of Lord Eddard Stark's wife, Tyrion manages to convince the tribes that therefore they share common enemies and are naturally on the same side.
In the books
In the A Song of Ice and Fire novels, the Eastern Road is much the same. The TV series largely omitted the appearance of the Bloody Gate, instead having Catelyn Stark's company encounter a patrol from the fortress led by Ser Vardis Egen.
In the novels, the names "Eastern Road" and "High Road" (over the mountains) have been used interchangeably. So far in the TV series, the road has only been referred to once in dialogue as "the Eastern Road" by Tyrion.