A one-way ANOVA is the simplest type of ANOVA analysis. You would use a one-way ANOVA if you are comparing one measurement across three or more groups; for example, marathon times across age groups. The "Descriptive Statistics" table gives helpful general statistics such as the number of observations, the mean and the standard error for each group. The "ANOVA" table give you your results, breaking down the ANOVA test. The final two columns, the F-statistic and the associated significance level, are your results. If the significance level is lower than your threshold value, often 0.05, then at least two of the groups are significantly different from each other.
You would use a two-way ANOVA when you have two predictive variables; for example, comparing marathon times across age groups and genders. The "Descriptive Statistics" table is more complicated, breaking down statistics by group and subgroup. The middle set of numbers on the "Tests of Between-Subjects Effects" table gives you your results -- a score and significance for each variable and a third score identified by both variable names connected by an asterisk. If this value is significant, it means that the variables interact, or that their effects are not independent. Significant interactions make interpretation more complicated.
The ANOVA results tell you that at least one group is different, but to figure out which one, you use a post-hoc test, usually Tukey or Bonferroni, which is displayed in the "Multiple Comparisons" table. This table gives the significance of the differences between each set of groups -- for example, people in their 40s vs. people in their 60s -- and both reports a significance value and marks the significant differences with an asterisk. If your two-way ANOVA showed a significant interaction, the results of the post-hoc tests might be misleading, and it is safest not to report these values.
The remaining outputs tell you whether your data meet the assumptions of data normality and equal variance among groups. If the significance value on the "Test of Homogeneity of Variances" table is less than 0.05, then your groups have unequal variances and your test results are suspect. You can evaluate the normality of your data on the box plot by checking that the whiskers are more or less the same size and that there are no outliers, represented by dots. Many datasets are not perfectly normal, but you will be able to tell whether you need to take your results with a grain of salt.