I have heard of a Maths journal having $A^*$, $A$, $B$ and $C$ rating, and have also heard of impact factor of $1.3$, $0.6, 0.33$, et-cetera.
Can someone please clarify me on what these two actually signify, and how you can use it to judge know the quality of articles in such journals?
Thanks in anticipation.
On
The impact factor of a journal is the average number of citations an article published in this journal received during two (or sometimes five) years after publication. This is a measure used mostly by administrative worked to judge the scientific output of a researcher (along with other numerical measures such as well known h-index, please google it if you did not here about it). It is very well known that this number has very little to do with the quality of the papers, you should carefully read this paper to see the details and very convincing data.
The rating A*, A, B, and C was developed by Australian Research Council to actually have something, which would be based on expert opinions and not on naked numbers, which as was very convincingly shown, can be easily manipulated. It was done only once due to the required resources, and I am not aware about new attempts to make a more up-to-date rating. More or less this rating reflects the opinion of the mathematical community and generally should be preferred over impact factor.
The impact factor tries to quantify the impact that the journal (i.e., the articles it publishes) have on the subject area relevant to the journal's aims. There are different ways of doing that of course, buy typically the number of citations made to articles is a significant component.
The letter indications of journals are a very rough attempt to grade the journal. This is usually done by having a team of experts in their fields assess the journals, rather than just following on some formula. Of course, this may (and often does) introduce biases. The current approach to these letter grades seems to be that it is largely meant for libraries when deciding on subscriptions. Some universities push (by offering cash bonuses for instance) their staff to publish in highly ranked journals, not always with good outcomes.
Typically one would judge the quality of a journal by one's own expertise, using the various number and letter grades associated to the journal as rough gauges along the way. Largely speaking, C ranked journals are not good to publish in. There are known cases where absolute nonsense was accepted to publication in certain such journals. Publishing in C ranked journals may actually have a negative effect on one's career.