Accuracy
Accuracy is the ability to produce correct sentences using correct grammar and vocabulary.
Accuracy is relative. A child in early primary isn't capable of the same level of accuracy as an adult.
Teachers who concentrate on accuracy help their students to produce grammatically correct written and spoken English.
Typical accuracy activities are: grammar presentations, gap-fill exercises, frame dialogues.
Fluency
Fluency is the ability to read, speak, or write easily, smoothly, and expressively. In other words, the speaker can read, understand and respond in a language clearly and concisely while relating meaning and context.
Fluency generally increases as learners progress from beginning to advanced readers and writers.
Language teachers who concentrate on fluency help their students to express themselves in fluent English. They pay more attention to meaning and context and are less concerned with grammatical errors.
Typical fluency activities are: role plays, speeches, communicative activities, games.
Accuracy vs Fluency
Taken as a given that students' needs should always dictate what you teach them, the question of whether it is more important to work on accuracy or fluency in the language classroom remains.
Many teachers believe that fluency is a goal worth striving towards only with students who are at a fairly advanced level. Other teachers, strong in the belief that the learning of a language is about communication, feel that fluency should be the main goal in their teaching and that it should be practiced right from the start.
More traditional teachers give accuracy paramount importance and test their students for accuracy and accuracy only!
Often a rigid educational system where tests and exams are the focus, will have students (and their traditional teachers) believe that language accuracy is what matters most, and giving the "correct" answers often becomes an obsession. Students who have been taught this way can complete any grammar gap-fill you care to give them, but will struggle to order a coffee in a real English speaking situation.
Tests and Exams
Excelling in tests represents but a small part of language competence and a lot of students' needs relate to performing in situations that are non-exam focused. However, English competence exams do drive a lot of the ELT industry and contribute to perpetuating the supremacy of accuracy over fluency.
For example, tests like IELTS and TOEFL are very much focused on accuracy, and so is FCE.
Benchmark testing instead is a great example of tests that are not accuracy based, though a higher degree of accuracy obviously accompanies a higher level of fluency.
PET or even CAE are also examples of tests that have a greater emphasis on fluency.
Methodologies in Practice
As far as teaching methodologies are concerned, very broadly speaking the communicative approach is the one that favours fluency the most, while the audio-lingual and grammar translation approaches favour accuracy.
Typically, at beginner level when the students don't have enough language to worry about fluency, teachers tend to focus on accuracy.
This carries on through to pre-intermediate level when fluency activities like discussions and debates are introduced.
At intermediate level, when the students are reasonably independent language users, a mix of accuracy and fluency is used, with the focus shifting to fluency as students advance.
To Sum Up
Students learning accuracy without fluency, and vice versa, is one of the biggest threats to successful learning and balancing accuracy and fluency should be the aim of any English language teacher.