Does "correlation" also mean the slope in regression analysis? I'm reading a paper and the author wrote:

The effect of A,B, C on Y was studied through the use of multiple regression analysis. A,B,C were entered into the regression equation with Y as the dependent variable. The analysis of variance is presented in Table 3.
  The effect of B on Y was significant, with B correlating .27 with Y.

English is not my mother tongue and I got really confused here.
First, he said he would run a regression analysis, then he showed us the analysis of variance. Why?
And then he wrote about the correlation coefficient, is that not from correlation analysis? Or this word could also be used to describe regression slope?
 A: 
First, he said he would run a regression analysis, then he showed us the analysis of variance. Why?

The analysis of variance table is a summary of part of the information you can get from regression. (What you may think of as an analysis of variance is a special case of regression. In either case you can partition the sums of squares into components that can be used to test various hypotheses, and this is called an analysis of variance table.)

And then he wrote about the correlation coefficient, is that not from correlation analysis? Or this word could also be used to describe regression slope?

The correlation is not the same thing as regression slope, but the two are related. However, unless they left a word (or perhaps several words) out, the pairwise correlation of B with Y doesn't tell you directly about the significance of the slope in the multiple regression. In a simple regression, the two are directly related, and such a relationship does hold. In multiple regression partial correlations are related to slopes in the corresponding way.
A: Analysis of variance (ANOVA) and regression are actually very similar (some would say they are the same thing).  
In Analysis of variance, typically you have some categories (groups) and a quantitative response variable.  You calculate the amount of overall error, the amount of error within a group and the amount of error between groups.
In regression, you don't necessarily have groups anymore, but you can still partition the amount of error into an overall error, the amount of error explained by your regression model and error unexplained by your regression model.  Regression models are often displayed using ANOVA tables and it's an easy way of seeing how much variation is explained by your model.
