What does it mean in if something isor not?
What does it mean in asset-pricing if something is priced or not?
It means does the market take risk factor X into account when setting the price of an asset. If it does then on average the asset will earn a Risk Premium over time. In the case of Beta it means do High Beta stocks earn a high long term return like the CAPM predicts (the CAPM says investors dislike Beta and only buy high beta stocks if they expect a bigger return, i.e. if they can buy them at a cheap price. We say "in the CAPM Beta is priced").
How do we determine empirically whether X is priced or not? We need long time series on a large number of stocks. We estimate the risk premia using the Fama Macbeth procedure and look at the statistical significance of the RPs (according to FM method). However, there is always some controversy because another choice of risk factors may be significant also, and a lot of these risk factors are related to each other in any case.... which are the "correct" ones? And the exact time period used affects the quality of the results also.
Note that there is a difference between something being priced and being a pricing factor. If it is priced, it just means that it has a positive risk premium; you are rewarded for taking the risk that corresponds to the factor. If it is a pricing factor, it is useful for pricing other assets. The difference is subtle, I recommend reading Cochrane: Asset Pricing for a detailed discussion.