As a teacher of child development I am familiar with several research experiments that show certain milestones in infants and toddlers. I did these with my three children when they were little because they were fun and informative.
#1 Newborn reflexes- Babies are born with, or soon develop, survival and primitive involuntary reflexes. Some are permanent and help us avoid danger (like the blink reflex) and others enable infants to practice later movements and fade away. See what your new baby can do: http://www.whattoexpect.com/baby-behavior/newborn-reflexes.aspx
#2 Object permanence- is the understanding that objects and people exist even when not visible. This is a major cognitive development for babies. Most infants can demonstrate this ability at about 9 months — so here’s what you do: Show your baby an attractive toy and while she’s watching, hide it under a blanket or pillow, if she looks for it, then she knows it still exists. If not, try again in a month or so.
#3 Self- recognition- the mirror or rouge test measures whether toddlers can recognize themselves. About half of 18 month olds and most of 24 month olds will react in a way that shows self-recognition. Here’s how you can try this: Put a dot of red rogue (or lipstick or paint) on your toddler’s nose and then stand him in front of a mirror. If your child touches or tries to wipe the mark off, it demonstrates awareness of how he’s “supposed to look” and will probably look embarrassed or avoid their reflection.
Have fun and remember that these tests do not take the place of professional screening or assessments (which you should seek if worried!)