A few quick answers:

• JAGS parameterizes the Normal distribution in terms of mean and precision (precision=1/variance), so a precision of 1e-5 means a variance of 1e5 or a standard deviation of 316. That this is "vague" or "weak" does depend on the scale of the covariate data. "Weak" essentially means that the standard deviation $\gg$ the scale of the data.
• I haven't read Best et al 1999 (as cited in your code), but Gamma(eps,eps) where eps << 1 is a typical weak prior for precisions: itgives a positive distribution with a large coefficient of variance (i.e., "vague") and a mean of 1 (JAGS parameterizes Gamma with shape and rate, so the mean is shape/rate = eps/eps = 1. This is again slightly sensitive to the scaling of the relevant covariate.
• You should be aware that the Gamma(eps,eps) (which is used in part because it's a conjugate prior for the precision of a Normal distribution, thus mathematically/computationally convenient) prior has been shown to have some bad properties in cases where the data is not very strong (and thus the prior has an effect); it often gives unrealistically large peak densities near zero, see e.g. Gelman 2006.

Gelman, Andrew. “Prior Distributions for Variance Parameters in Hierarchical Models.” Bayesian Analysis 1, no. 3 (xx xx 2006): 515–33.

A few quick answers:

• JAGS parameterizes the Normal distribution in terms of mean and precision (precision=1/variance), so a precision of 1e-5 means a variance of 1e5 or a standard deviation of 316. That this is "vague" or "weak" does depend on the scale of the covariate data. "Weak" essentially means that the standard deviation $\gg$ the scale of the data.
• I haven't read Best et al 1999 (as cited in your code), but Gamma(eps,eps) where eps << 1 is a typical weak prior for precisions: it gives a positive distribution with a large coefficient of variance (i.e., "vague") and a mean of 1 (JAGS parameterizes Gamma with shape and rate, so the mean is shape/rate = eps/eps = 1. This is again slightly sensitive to the scaling of the relevant covariate.
• You should be aware that the Gamma(eps,eps) (which is used in part because it's a conjugate prior for the precision of a Normal distribution, thus mathematically/computationally convenient) prior has been shown to have some bad properties in cases where the data is not very strong (and thus the prior has an effect); it often gives unrealistically large peak densities near zero, see e.g. Gelman 2006.

Gelman, Andrew. “Prior Distributions for Variance Parameters in Hierarchical Models.” Bayesian Analysis 1, no. 3 (xx xx 2006): 515–33.

A few quick answers:

• JAGS parameterizes the Normal distribution in terms of mean and precision (precision=1/variance), so a precision of 1e-5 means a variance of 1e5 or a standard deviation of 3300. That this is "vague" or "weak" does depend on the scale of the covariate data. "Weak" essentially means that the standard deviation $\gg$ the scale of the data.
• I haven't read Best et al 1999 (as cited in your code), but Gamma(eps,eps) where eps << 1 is a typical weak prior for precisions: itgives a positive distribution with a large coefficient of variance (i.e., "vague") and a mean of 1 (JAGS parameterizes Gamma with shape and rate, so the mean is shape/rate = eps/eps = 1. This is again slightly sensitive to the scaling of the relevant covariate.
• You should be aware that the Gamma(eps,eps) (which is used in part because it's a conjugate prior for the precision of a Normal distribution, thus mathematically/computationally convenient) prior has been shown to have some bad properties in cases where the data is not very strong (and thus the prior has an effect); it often gives unrealistically large peak densities near zero, see e.g. Gelman 2006.

Gelman, Andrew. “Prior Distributions for Variance Parameters in Hierarchical Models.” Bayesian Analysis 1, no. 3 (xx xx 2006): 515–33.

A few quick answers:

• JAGS parameterizes the Normal distribution in terms of mean and precision (precision=1/variance), so a precision of 1e-5 means a variance of 1e5 or a standard deviation of 316. That this is "vague" or "weak" does depend on the scale of the covariate data. "Weak" essentially means that the standard deviation $\gg$ the scale of the data.
• I haven't read Best et al 1999 (as cited in your code), but Gamma(eps,eps) where eps << 1 is a typical weak prior for precisions: itgives a positive distribution with a large coefficient of variance (i.e., "vague") and a mean of 1 (JAGS parameterizes Gamma with shape and rate, so the mean is shape/rate = eps/eps = 1. This is again slightly sensitive to the scaling of the relevant covariate.
• You should be aware that the Gamma(eps,eps) (which is used in part because it's a conjugate prior for the precision of a Normal distribution, thus mathematically/computationally convenient) prior has been shown to have some bad properties in cases where the data is not very strong (and thus the prior has an effect); it often gives unrealistically large peak densities near zero, see e.g. Gelman 2006.

Gelman, Andrew. “Prior Distributions for Variance Parameters in Hierarchical Models.” Bayesian Analysis 1, no. 3 (xx xx 2006): 515–33.