TL;DR version: Zhuge Liang was a great politician and civil administrator responsible for building and maintaining the Shu state. He was, however, a terrible military leader and had a mixed military record.

My answer in another thread:

The scholarly consensus on Zhuge Liang is that he was a great civil administrator and politician, and was one of the principal figures responsible for the construction of the Shu Han state. As a strategist, he wasn't that great and his military record was mixed. For example, in a scholarly hindsight, Wei Yan's plan of attacking Chang'an through a lesser known route would have had a far greater chance of success, but Zhuge Liang shot his plan down (leading to Wei Yan's disillusion) because he considered it too dangerous. The result was that all the Northern Expeditions failed in their strategic objectives, as Shu Han lacked the manpower and resources to wage a prolonged campaign.

But the question is, why is Zhuge Liang popular? Well, there are several reasons:

• The political atmosphere changed post-Jin when Cao Wei was considered the legitimate successor to the Han. In later periods, Liu Bei was seen as the rightful heir to the Han after Cao Pi deposed the last emperor. Zhuge Liang, as one of the principal advisers of Liu Bei and being responsible for the creation of the Shu Han state, was seen in a very positive light.

• Zhuge Liang not only predicted the Three Kingdoms, he was also responsible for it. He laid out the strategic plan that would lead to the creation of Shu Han in the southwest and formed an alliance with Eastern Wu against Cao Cao, leading to the Three Kingdoms. This is the famous Long Zhong Dui (隆中對).

• After Liu Bei's death, Zhuge Liang did not seize power for himself and kept his promise to assist Liu Bei's son (loyalty was very important in Chinese culture).

• Zhuge Liang launched the Northern Expeditions, where the Shu won some military engagements, most notably from Zhuge Liang's use of ambushes, despite the campaigns being overall failures. As commander-in-chief, it's natural for him to get a lot of the credit, and this was later exaggerated in Romance.

• Zhuge Liang opposed Liu Bei's punitive expedition against Wu, which ended in the disastrous Battle of Xiaoting.

• He was primarily responsible for bringing in a number of major Shu military figures: Jiang Wei, Pang Tong, Ma Su chief among them.

These are all achievements of Zhuge Liang and some, such as the Long Zhong Dui, are of great importance. As the underdog of the Three Kingdoms, Shu launched six expeditions under Zhuge Liang against Wei and put Wei on the defensive, a feat that most would consider highly impossible given Shu's disadvantages. This was only possible because of Zhuge Liang, so he was held in high regard by later generations. Myths and oral tales of the Three Kingdoms circulated long before Romance, and in writing Romance, Luo Guanzhong probably absorbed a lot of those and then used his own imagination to paint Zhuge Liang as a great strategist. Going back to what I said about Romance, it was really a tale of righteousness, justice, and morality. Liu Bei was the epitome of righteousness and so his principle adviser then had to be someone of great talent.

It's worth noting that Zhuge Liang only rose to military prominence (in actual history) after most of the other (and arguably better) Shu strategists were already removed from the scene: Xu Shu defected to Wei, Pang Tong died in battle, and Fa Zheng died of natural causes. Xu Shu masterminded a lot of Liu Bei's early, albeit unsuccessful, strategies against Cao Cao and recommended Zhuge Liang to Liu Bei in the first place. Pang Tong organized the invasion of Ba province and accompanied Liu Bei into battle, and Fa Zheng orchestrated the critical victory at Hanzhong. Prior to Liu Bei's death, Zhuge Liang was not a regular member of the campaigns. He took a more backseat role, governing Liu Bei's territories, administering his household, and maintaining supply routes.

The real rise of Zhuge Liang only occurs after the death of the Shu old guard. Guan Yu was executed, Zhang Fei was assassinated, Fa Zheng, Ma Chao, and Mi Zhu all died of natural causes, and Huang Zhong was killed in battle. The only "great" Shu generals left were Zhao Yun (already old), Wei Yan, Jiang Wei, Wang Ping, and Ma Dai. As a result of this, Zhuge Liang was chosen to command the army in suppressing the Nanman rebels. In Romance, Zhuge Liang performs a lot of tactical feats and maneuvers, capturing Meng Huo seven times, but in actual history, they simply moved in, subjugated the rebels, and moved out. For the Shu, this was considered a victory, but tactically it didn't really change anything and wasn't considered that important. But with this victory, and as the architect of Shu's alliance with Wu, Zhuge Liang became one of the few viable figures left to command the northern expeditions.

Zhuge Liang does deserve a lot of credit for rebuilding the Shu military and keeping the state together in the aftermath of Xiaoting, where the majority of the Shu army was destroyed. He caused the defection of a number of Wei generals, most notably Jiang Wei. But as a commander, it's important to remember that he lost nearly every single major engagement of the Northern Expeditions. When he didn't lose (such as at Jianwei or Wuzhang), the result was often inconclusive as he could not capitalize on his victories. All territorial gains he made were later reversed. His only real result was that the expeditions led to the death of many important Wei generals: Zhang He and Wang Shuang were killed in battle, while Cao Zhen and Hao Zhao died of illness. So in reality, the only real successful campaign you see from Zhuge Liang is the suppression of the Nanman rebels.

Sources used: Zi Zhi Tong Jian by Sima Guang and translated by Rafe De Crespigny and Achilles Fang.

https://openresearch-repository.anu.edu.au/html/1885/42048/index.html

http://the-scholars.com/viewtopic.php?f=23&t=22087

Zhuge Liang was one of the most talented man of his era.

He presented the LongZhongDui to Liu Bei at Jing Province when Liu Bei was simply a retainer of Liu Biao. During the meeting, he foresaw the eventual tripartite of China, outlining the strengths and weaknesses of Cao Cao, Sun Quan, Liu Biao, Liu Zhang and Zhang Lu. He then encouraged Liu Bei to form a base at Yi and Jing Province and ally with Sun Quan to attack Cao Cao in hopes of restoring China under proper Han rule.

He was chosen to be a diplomat to persuade Sun Quan to ally with the Lius to fight against Cao Cao. He did his job remarkably well considering the time constraints and political tensions within Sun Quan regime.

When Liu Qi, the son of Liu Biao engaged in a succession stuggle with Liu Cong, it was Zhuge Liang who outlined the precarious situation for him and encourage him to leave for Xiakou to escape the danger in Jing. This would later enable Liu Bei along with Liu Qi and Sun Quan to make a last stand against Cao Cao at Chibi.

It is noted that whenever Liu Bei went to war, Zhuge Liang would remain at base to supply provisions and collect taxes, ensuring that the various retainers are doing their jobs properly and recommending talents like Pang Tong to Liu Bei.

When Liu Bei was isolated at Yi province and demanded reinforcements from Jing, Zhuge Liang was the coordinator of the relief forces.

Again, during HanZhong, he was the one in charge of reinforcements, supplies and good governance.

When Jiang Wan, future regent of Shu Han was to be punished for improper governance, it was Zhuge Liang who noticed his talents and intervened to safe Jiang Wan.

After the disastrous Yiling campaign by Liu Bei, he would later die at Baidicheng, entrusting his Empire of Shu Han to Zhuge Liang and Li Yan.

For the next two decades or so, Zhuge Liang would pacify the South, reaffirm an alliance with Sun Quan and marched 5 times against Cao Wei.

Though his Northen Expeditions were repeated failures, it is important to note that Shu Han was inferior to Cao Wei in terms of resources, manpower and talents. Yet Zhuge Liang was able to annex two commanderies, kill Zhang He and keep Cao Wei on the defensive and off guard again and again.

In addition to killing Zhang He, the last remaining of the Cao Wei Five Elite Generals, he was also able to win a couple of battles and relocated many populace of Cao Wei to Shu Han.

He would also make use of various personal inventions to aid in his expeditions.

Zhuge Liang was well respected by the commoners and the various ministers and generals of Shu Han.

Zhang Yi notes

"His Excellency does not leave out those afar when he deals out rewards, nor does he show partiality to the near when he metes out punishments. Rank cannot be obtained when one does not deserve it, nor can one escape punishment by dint of powerful influence. This is why all, regardless of wise or foolish, forget their own selves in serving him." 

Furthermore, Zhuge Liang popularity was such that when Zhuge Liang passed away, many commoners wanted to erect shrines everywhere and even offered private sacrifices to him year after year.

Zhuge Liang was held in high esteem by even his rivals. Jia Xu and Liu Ye praised his administrative ability, Sima Yi called him a genius and when his death reached the State of Sun Wu, the various people of Sun Wu thought that Shu Han will eventually fall to Cao Wei as a result of Zhuge Liang death. They thus made preparations by stationing more troops on their western border that is aligned with Shu Han.

This is the edict by Liu Shan, second Emperor of Shu Han.

The imperial edict read: 'You have embodied the civil and military qualities, abundantly possessing insight and wisdom. You have received from the Late Emperor the charge of an orphan, and have supported me. You have made the broken line to continue and the declining to flourish. Your aim has been to tranquilize the disturbances. You have put the Six Armies in order; there has not been a single year when there was no campaign. Your divinely martial fame has become brilliant and your prowess has shaken eight domains. You were about to accomplish extraordinary work in these latter days of our Han, and to share the great achievements of Yi Yin and the Duke of Zhou. Alas, when things had come to such a crux as to be reaching success, you fell ill and are lost to me. I am as desolate as if my liver and heart were torn asunder.

Now, to honor virtue and set forth achievements, to record deeds and designate canonization, is to make them illumine the future and to write them down beyond decay. I herewith send the zuo zhonglangjiang Du Qiong to carry the Plenipotentiary Tally and confer on you the seal of chengxiang and Lord of Wuxiang and canonize you as 'Loyal and Martial' Lord. If the soul is cognizant, take this especial grace commendably. Alas! Alas!'”

And these are appraisals by Chen Shou

Chen Shou's "Comment" says:  As chancellor of the state, Zhuge Liang soothed the people, demonstrated rules and regulations, simplified the official hierarchy, followed compromise measures, opened up his sincere heart, and spread fairness and justice. Those who served loyally and assiduously he would always reward even if they might be his enemies; those who violated laws or were remiss in their duties, he would always punish even in they were his relatives. Those who recognized their guilt and apologized, he would always pardon even if their misdeeds were heavy; those who cunningly prevaricated and elaborated, he would always put to death even if their misdeeds were light. There were no good deeds, however trifling, that he did not reward; there were no wicked deeds, however small, that he did not punish. He was well versed in the manifold affairs and took basic measures; he demanded reality corresponding to the name, and did not appreciate falsity. In the end, all within the land stood in awe of him and loved him. His punishments were indeed harsh and strict, yet there were none who complained; this was because he was equitable and fair in mind and clear in his exhortations. He could be said to be a man of outstanding ability who knew how to govern, a peer of Guan Zhong and Xiao He. Nevertheless, he made campaigns every year without achieving result. It would seem that resourceful generalship in response to changing situations was not his forte." 

And Xi Zuo Chi

In ancient times, Guan Zhong took the city of Pian, with three hundred families, from the chief of the Bo family, yet the latter did not utter a murmur to the end of his life. The sage (i.e., Confucius) thought this not easy. Zhuge Liang's causing Liao Li to weep and Li Ping to die are not mere cases of not uttering a murmur. Now, water is as level as anything can be, yet the wicked take their warning from it; a mirror is as perspicacious as can be, yet an ugly person has no anger when he looks into it. The reason why water and mirror probe into the depth of things, and yet cause no murmur, is that they are impartial. Being impartial, water and mirror escape slander. Then how much more is it so with a great and superior man, who cherishes the desire to let live and spreads the virtue of pity; whose laws are applied because they cannot but be applied, and whose punishments are meted out to those who have transgressed of themselves; who confers ranks but not out of favoritism, and punishes to death without showing anger? Can there be any one in the world who does not submit to him? Thus can Zhuge Liang be called one who knows how to mete out punishments; there never has been such a one since Qin and Han.