The parameter Beta0 has the same units as the Y values (the outcome variable).
The other best-fit parameters have the units of the Y variable divided by the units of the corresponding X variable.
Consider this example model again.
Blood pressure = Y = Beta0 + Beta1*age +Beta2*weight +Beta3*gender + random scatter
Beta0 is expressed in units of the Y variable, which is mmHg. It is the predicted value of Y when all X variables equal zero. For this example (and many others) this is a bit silly, as it would be the average blood pressure of men (since gender is coded as zero) with age=0 and weight=0! Better just to think of it as a constant in the model.
If blood pressure is measured in mmHg and age is measured in years, the variable Beta1 will have units of mmHg/year. It is the amount by which blood pressure increases, on average, for every year increase in age, after correcting for differences in gender and weight.
If weight is measured in kg, then Beta2 has units of mmHg/kg. It is the average amount by which blood pressure increases for every kg increase in weight, adjusting for differences in age and gender.
Gender is a dummy variable with no units, coded so that males are 0 and females are 1. Therefore, Beta3 has units of mmHg. It is the average difference in blood pressure between men and women, after taking into account for differences in age and weight.
The only way you could really know the best-fit values of the parameters in the model would be to collect an infinite amount of data. Since you can't do this, the best-fit values reported by Prism are influenced, in part, by random variability in picking subjects. Prism reports this uncertainty as a 95% confidence interval for each parameter. These take into account the number of subjects in your study, as well as the scatter of your data from the predictions of the model. If the assumptions of the analysis are true, you can be 95% sure the true best-fit value of the parameter lies within that confidence interval.
Prism also presents the standard error of each parameter in the model. These are hard to interpret, but are used to compute the 95% confidence intervals for each coefficient. Prism shows them so that its results can be compared to those of other programs.