Hi anynamesleft
Mumsnet Learning pages (supported by Pearson) has a good section on Assessment and NC Levels here: www.mumsnet.com/learning/assessment/national-curriculum-levels
What the levels mean: www.mumsnet.com/learning/assessment/what-national-curriculum-levels-mean
How a child should progress through the levels: www.mumsnet.com/learning/assessment/progress-through-national-curriculum-levels
On top of this some schools also use APP (Assessment of Pupil Progress), which basically breaks down progress in reading, writing and maths by observation of working to a number of AF (Assessment Focus) levels. The idea is that by observing work at specific points - the teacher can better track progress.
Example from Hertfordshire CC here for reading/ writing: <a class="break-all" href="http://www.thegrid.org.uk/learning/english/ks3-4-5/ks3/assessment/index.shtml#app" rel="nofollow" target="_blank">www.thegrid.org.uk/learning/english/ks3-4-5/ks3/assessment/index.shtml#app (Please note the grids for reading & writing go up to NC Level 8 (which is secondary school levels) and primary is only required to Level 4, but can go to Level 5 and exceptionally goes up to about Level 6 - but you can clearly see the differences in attainment across a number of different indicators (AF levels)).
Example for maths as 'I can statements' here: www.usethekey.org.uk/popular-articles/2018i-can2019-statements-for-subject-specific-skills-at-key-stages-1-and-2#section-6
Basically most LEAs are working to some form of this. Some schools apply this to all pupils whilst others apply it to a sub-set of pupils and then use these pupils to benchmark levelling for the remainder of pupils in a class.
Why are you being given levels now? Well OFSTED is requiring that schools inform parents of progress and that pupils clearly understand where they are and what they need to do to improve. Many schools have studiously avoided discussing levelling of pupils - because most parents will naturally want their children to be doing well and difficulties might arise if they aren't. Our school only introduced a simplified parent-/ student-friendly version of APP against NC Levels in the run-up to OFSTED inspection. Personally I've found it very useful as my DD1 was seriously struggling in reading, writing & maths in KS1 (largely because she learns through doing and there was a strict little or no homework policy & because 'average' students got a lot less direct time with the teacher for things like guided reading or maths group work). I've posted elsewhere about our solutions.
SATs testing KS1: this is now largely carried out through teacher assessment with very little formal testing. In most cases the child is unaware this is going on, but they may be selected out individually or in small groups to do something like a reading comprehension 'exercise', which is in fact a test. They will not be sat at a desk for an hour filling in some form of standardised test.
SATs testing KS2: is more serious and children can get wound up but this is really about the teachers/ school being anxious to do well because these results are reported and used to rank schools. So in fact the pressure is on the school to have got pupils to a certain standard - rather than on the pupil. For example, it's not their fault if they don't know how to divide, because it wasn't taught - as can be the case at our school. What is important for the pupil is these results are what the next school (the secondary school) will first know about your child. So they will initially be placed into groups for subjects on the basis of these results (so if your child gets L3, they'll be placed in a group with other pupils who are struggling and if your child gets L5 they'll be placed in a group with other pupils working at above average levels). I hasten to add that most secondary schools also do their own testing in the first term to ensure that children are placed into the appropriate set when starting secondary school.
I've posted elsewhere that the way to view these national tests is that taxpayers (with or without children) are funding education and they are entitled to understand that the schools are performing to a certain standard. When you look into it the NC Levels expected (2b at end KS1 (Y2) and 4b for KS2 or end of Y6) they really aren't that demanding.
HTH