The word Guru means one who dispels darkness – a teacher.
The word Acharya means one who teaches the right conduct – again a teacher, basically.
But other than the straight meaning, in Hinduism both Guru as well as Acharya are associated with a master to whom a religious/ spiritual aspirant goes and seeks guidance in the matter of divine/ spiritual/ adyatma knowledge.
In certain sampradayas (traditions) (e.g. Vaishnava), the master is referred to as Acharya rather than Guru.
Again there are finer differences.
Normal Gurus may or may not have the divine authority to teach spirituality to others, but they may act as guru based on appointment as a spiritual heirs of their gurus, or by the strength of whatever spiritual experience or workable knowledge they have gained through experience. Whatever they have gained or experienced through the path or technique they followed or received from the blessings of their Gurus, they will teach to their followers and it could be limited and narrow too (example: worshiping a specific God form only, instructing a specific mantra only, teaching a specific pranayama method only, instructing a particular meditative method only, vouching on a particular religious path and discipline only).
The normal Guru is likely to spread his knowledge as well as his ignorance to his followers (e.g. refusing to acknowledge or approve any other path, any other God forms, any other sadhana technique etc; he may even criticize and condemn others). (Please note: Not all Gurus may behave like this). A normal Guru may also be very possessive about his cult and followers and if any of them leave in search of anything better, they may condemn and criticize them. (Please note: Not all Gurus may behave like this).
In some cases, a guru may not be very learned in scriptures, but he may have personally experienced God. His experience could be anywhere from getting a glimpse of God/divinity, a sighting of light to the highest knowledge, to the highest state of ‘sahaja nirvikalpa samadhi‘ — remaining permanently immersed in the state of unity with God without any duality. A guru of the highest state of personal experience/ attainment is known as Sadguru. (Explained at the end).
At the elite level, an Acharya is the one who, by deeply learning scriptures from competent Gurus and through dedicated self-study, analysis, understanding, tapas (practicing austerities), prayers, meditation and so on gets a very clear idea about a particular school of thought or philosophy that he feels extremely sure of. Then he goes about (by divine will) to teach his school of thought to ardent seekers and followers. He engages himself in intellectual arguments with pundits of different other schools and establishes supremacy of his ideas. (Ex: Sankaracharya, Ramanujacharya, Madhvacharya).
On the other hand, on a normal practical level, an Acharya is invariably quite learned (intellectually) in scriptures and most probably with adequate Sanskrit knowledge. He might have gained quite some sound theoretical knowledge about a particular sect, school of philosophy, Worship of god-form, a specific scripture etc .
An acharya, may or may not have had any personal experience of God or divinity; yet, by virtue of certain sampradayas (traditions) an Acharya may get appointed as the chief of a Mutt by purely the strength of his scriptural knowledge and capacity to lecture. He may take up sanyas by leaving behind his erstrwhile grihasthasrama (house holder life) in order to don that post. (e.g. South Indian Vaishnava Mutts like Srimad Andavan Ashram or Jeeyar).
There may also be certain traditions where only a Brahmachari (unmarried) sanyasin will be appointed as the head/ future head i.e. Acharya of the mutt and not Grihasthas. (e.g. Sankara Mutt).
Acharyas of formal Mutts will strictly teach their sampradaya only to their followers. Depending on their spiritual attainment (if and when they get it) they may or may not acknowledge other paths and methods as valid or acceptable, but they will be strictly bound by their tradition and will not recommend or suggest or openly acknowledge any other method, practice or path as a valid one.
In certain mutts (example: Ramakrishna mutt), there may be several sanyasins, but only some will be authorized act as gurus and acharyas. Those who are recognized as Gurus alone will be authorized to give Mantra diksha to their disciples. Those who are recognized as acharyas will be the ones who have acquired good scriptural knowledge and they will be the ones to conduct scriptural classes, give discourses and lectures etc. Some acharyas may evolve into diksha Gurus too.
A SADGURU (Satguru)
Sadguru indeed is the Guru of the highest order; he/she is the one who has experienced divinity in full and remain ever immersed in it. In the path of bhakti he/she will be the one who has attained prema bhakti and attained union with his personal God. In the path of Gnana, he/she will be the knower of Self , the one who has personally experienced the state of Advaita. In the path of yoga, he/she is the one who is established in Sahaja Nirvikalpa Samadhi. A satguru is one in whom the mind has been annihilated. He/she is the one who has totally eradicated ego; he/she is totally freed of vasanas. He/she is a jeevan mukta (liberated while being alive).
A satguru has reached the peak and from the peak, he/she sees that all the paths are leading to just one end. He/she is not bound by any specific sampradaya through which he may have attained the highest state. He/she acknowledges and accepts all the valid paths, gurus, scriptures, techniques and his only interest is to guide people from any path, any sampradaya to follow their path with true understanding and attain God. He/she is not compartmentalized. He/she accepts one and all. He/she goes even beyond boundaries of religion and attracts people from all religions. Love for mankind flows unconditionally from their heart.
A satguru may not really appear in any Guru or Acharya Parampara. A Satguru in all probability will be a swayambu (self existing). A Satguru may be born in a remote village to simple and unassuming parents who have least to exposure to anything divine. A Satguru could be an avatara purusha. He/she arrives by divine will to teach the mankind. To show a new path. To cleanse a stagnant and dirtied religion or a religious sect. He/she is like a large steamer that can carry hundreds of ardent devotees and disciples across the sea of Samsara.
When a Sadguru or Avatara Purusha comes, their basic mission is to elevate humankind spiritually. Yet, they may also act in the worldly plane in providing means for common people to meet their essential worldly needs, and in undertaking large scale socially beneficial acts of philanthropy. They may engage themselves in activities like building educational institutions, hospitals, home for destitutes and so on. They may actively engage themselves in relief works after natural disasters. Further, they are also capable of using their extraordinary spiritual powers in helping individual devotees to overcome griefs and sufferings in their personal lives like severe diseases, extreme poverty, emotional problems, addictions, depression and so on. This way, they tend to elevate people’s focus from the mundane issues to more spiritually progressive way of life.
A satguru may come first and a Guru-Sishya Parampara may get established after him. The parampara of his/her main sishyas (future gurus) will spread his light to more and more people. The guru parampara may run for hundreds of years; it may or may not produce any more satgurus in this tradition. It may also get deteriorated and decimated over a period of time.
Then comes another avatara, another satguru at some other location, at some other context to re-establish dharma suited to the times. It keeps happening so in India from time immemorial. That is the greatness of Bharat and the Sanatana Dharma – Hinduism.