Documentation for JIRA 4.1. Documentation for other versions of JIRA is available too. 
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In Atlassian's support JIRA, when a user creates an issue with "Critical" priority, it will display the "Priority Explanation" field.
This can be achieved by performing the following steps:
<script type="text/javascript">
priority = document.getElementById('priority');
if (priority) {
target = document.getElementById('customfield_10420FieldArea');
// Hide the target field if priority isn't critical
if (priority.value != 2) target.style.display='none';
priority.onchange=function() {
if (this.value == 2) {
target.style.display = '';
} else {
target.style.display='none';
}
}
}
</script>