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I'm dealing with hierarchical model where $Y_i$ are from normal distribution. About variance the formulation is the following: Similarly, the data contain substantial information about the measurement error, $\sigma^2$, so the relatively noninformative Jeffreys' prior, $l/\sigma^2$, was used for this parameter So, what exactly prior should I chose? Could you give some practical reasons so that I won't need to read lots of literature about Jeffrey's prior?

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Those two are my favorite questions about the subject:

Weakly informative prior distributions for scale parameters

What is an "uninformative prior"? Can we ever have one with truly no information?

Also, it might help taking a look at section 2.9 of:

Gelman, Carlin, Stern and Rubin (2004), Bayesian Data Analysis.

And also Gelman's paper mentioned in @Michael Chernick answer:

A. Gelman (2006), Prior distributions for variance parameters in hierarchical models, Bayesian Analysis, vol. 1, no. 3, pp. 515–533.

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