There are two types of ages: the physical and emotional age. A childish adult is basically a person whose physical age is that of an adult, but whose emotional age is that of a child.
Looking for Who to Blame
When an adult hardly takes responsibility for his/her actions, and constantly looks for who to blame, that's might be a sign of emotional immaturity in an adult. Typically, when things go wrong children look for who to blame, while adults take responsibility and try to figure out how to fix the problem.
Poor Impulse Control
Children tend to be impulsive when they are hurt or mad - they speak recklessly or take impulsive action without pausing to think about the consequences. Adults that mimic such behavior when reacting to issues show signs of emotional immaturity. Adults are typically better at self-control and understand themselves enough to calm themselves and resist the impulse to act before they think in a given situation. This is not to say that adults don't occasional have 'slips', but this happen less with an adult than a child.
Immature Defense Mechanisms
One primitive defense mechanism is denial: 'I didn't say that!'; 'I never did that!', when the individual actually said so. These are immature defense mechanisms used by a child to protect himself/herself or escape tight corners. Another primitive defense mechanism, is attacking or being hostile to anyone who says something different or has a different view from yours. Adults that have such immature defense mechanisms can to a large extent be considered childish adults. They still have a lot of growing up to do in this aspect.
Need to be Center of Attention
This point speaks for itself. It's safe to say that an adult whose priority is to always be the center of attention still has some growing up to do.
Lack of an 'Observing Ego'
Emotionally mature adults have outbursts, but what's different about them is that after their outburst, they soon after realize, with their 'observing ego', that their outburst was inappropriate. They can see with hindsight and admit that such behavior was against their value system. On the other hand, children have not yet been able to internalize guidelines for respectful behavior and develop a value system. As a result, they don't have the ability to judge when their actions are out of line and thus blame others. Adults whose behaviors are in line with this child like response can be considered childish adults.