Students should be assessed in order to ensure they are getting a quality education. Assessments help teachers measure a student's progress through the course as well as their understanding and knowledge of the subject. Assessments also help teachers improve their own teaching practices. Students can show their knowledge and prove their mastery of particular subject matter through these assessments. Without them, it would be difficult to gage how much a student has learned and if they are ready to build on what they have learned.
Something that I hear a lot in our professional development is "assessment of learning and assesment for learning." Both are important, but I think many people forget about the latter. Of course, we need to assess what the students have learned; however, we also need to use assessment to help them learn more. When a student takes a formative assessment, however, and is taught how to look at that assessment and understand what he/she learned and still needs to work on, it can help that student take control of his/her own learning and make it more meaningful.
Another reason why assessment is important is that it allows the teacher to also take measure of what has worked and not worked. In other words, assessment tests give vital feedback to teachers about their own educational methods and skills. Of course, students might not be doing the work, but assessment tests also can reflect a teacher's pedagogical method, especially if these assessments are compared.
I like the word "assessment" over "test" personally, because the foundational purpose for assessment is to understand what a student has learned as a result of a given lesson. Essentially, it is a mode for discerning whether the objectives of a lesson have been met. Assessment, at its core, is just as much to see whether the teacher and the lesson successfully communicated the knowledge or skills to the students.
We all need to be assessed on our performance to see what progress we have made towards a certain goal. Our role as teachers is to educate students to become knowledgeable and responsible citizens. Whatever area of the curriculum we deliver, it is important that both teacher and student are aware of their starting position in terms of knowledge and understanding, and the shift in position after further teaching and study. As others have said, there are a myriad of good (and bad) assessment procedures, but if we do not assess we do not equip the students with an understanding of their abilities, nor do we establish the validity and efficacy of the programmes we implement.
Assessment is the only way a teacher can find out what a student has learned. That being said, there are multiple ways to assess students--one does not always need to use tests/exams to assess. Simple discussions work as assessments as well.
A lot of assessment is simply traditional testing, and some types of assessment are less effective than others. But testing as a means of assessment is traditional for a reason, as it is a motivator for students to apply themselves. Sometimes, I find it is the only motivator. So besides giving important feedback to the student about their progress, and giving some measure of that progress that parents can see, it also affects the approach students take to our classes and subject matter.
Assessment is vital because it helps both the teacher and the learner to know if they have learnt anything, which then allows the teacher to identify any gaps in the learner's knowledge and support that learner to reach the intended learning outcomes. Without assessment, education is nothing more than a teacher standing in front of a class and hoping that his or her students magically learn something.
Students should be assessed because there is no other way to find out whether they have learned anything.
Of course, assessments do not just have to be pen and paper tests. They can be done in a number of ways. Regardless of how they are done, they must be done. Much of the purpose of education is for students to learn things. If we do not assess them, we do not know if they have learned. If we do not know if they have learned, we do not know whether we have done our jobs properly and we do not know if they know the things that they will need in order to succeed as they go along in school and in life.