It is the first question on any student's mind before even starting to prepare for GRE.. "What is the average GRE score!" It helps them to judge their performance or a score based on that. ETS is quite aware of this query and they tried to address it by releasing an updated version of their Snapshot Report. So currently standing in 2014 the Average GRE score according to the data provided by ETS is 302.2; 150.8 for verbal and 151.4 for math. And the AWA average is 3.7.

Is the average GRE score important?

Not really.. .Yes i know its surprising. The averages above were calculated based on results from over 450,000 students who took the GRE between August 2011 and June 2012. (It’s now 2014 – things change!) This is a diverse group of students, applying to many different programs with many different requirements. As you’re applying to graduate programs, the average score of all test takers is irrelevant. For you, the average GRE score for your particular program at your target school is much more important — this is the score you need to exceed.
For example, students in the top 20 Engineering colleges in the US have an average GRE Score of 320.So you will obviously want to score higher than 320.
In their snapshot, ETS also provide average scores by intended graduate major:

Average Scores
Intended Major	Quant	Verbal
Business	152	149.5
Education	147.3	150.8
Engineering	158.9	148.7
Humanities & Arts	149.2	156.5
Natural Sciences	153.1	150.8
Social Sciences	150.1	152.9
Other Fields	148.9	149.7
Undecided	152.6	148.3
No Response	151.4	148
Overall	145.8	150.8

The table above provides a helpful baseline, but note that these are average scores for all schools. It is highly recommend to do research for the specific colleges and universities that you are targeting. You can find it in The US News & World Report.

What is the score you should target?

There is no straightforward answer to this question. It depends on many things. Too be idealistic you should always target for a perfect 340 score. Anyways jokes apart, there are a few things you can check before you can choose a target score.
1. Check the average score in your intended major(list given above)
2. Make a list of colleges & universities you want to apply.
3. Go to The US News & World Report and find out the average GRE score of your college or university.

Lastly you must understand that GRE score is not everything in your application. Its only a part of it, so do not worry if your GRE score is not much high than the average or even lower than that. If you are confident on the rest of your profile you should go for it. Statistically speaking the universities take a lot of people below their official average GRE score, that's why it called average.
I hope this helps! Enjoy!