Formative and summative assessment are both very important terms for teachers, as they are two very different types of assessment. Assessing for teachers is all about finding out what students know: how much they have learnt what the teacher is trying to teach them and how well. The difference is that formative assessment is a term used to refer to any type of assessment that the students receive in class to check understanding. Formative assessment might include a quiz or a mock exam, for example. Summative assessment is used to refer to any final exam that the students are preparing for. So, an example of a summative assessment would be finals, or end of year exams, or the delivery of a big project.
Formative assessment normally focuses on smaller chunks or units of learning than a summative assessment, which is normally broader in scope. In Britain, in recent years there has been a massive focus on formative assessment in terms of making sure students have learnt something before they leave the classroom, rather than waiting until they have left and finding out when they do badly in the exam that is a month or two away. Every class then is expected to display some kind of formative assessment, though this can be done in a variety of different ways.