Gagging can mean many things...
I got an email from a mom about her son's picky eating and how he often gagged when given new foods (he is a young toddler).
Dealing with these questions over email is difficult. Gagging can mean so many things, from a normal physiological reflex, to more rarely a sign of a more serious problem.
Occasional gagging in infants and young toddlers is normal. As children learn to eat, they have a very strong gag reflex that will weaken (and move farther back in the mouth) over time. Gagging is very common for the older infant learning to eat foods with more texture. Children gag to protect their airway.
For a child who has eaten well, grown well and is healthy, it can be a little disconcerting, but is most often not cause for concern. Infants like to mouth toys and even gag themselves (my daughter went through a phase of gagging herself with toys). While this looks odd, they are doing important work to prepare them for eating solid foods. As much as possible, try not to react with too much concern. If you hop up and hold a cloth under her, or appear panicked every time she gags, she will feel scared and this may upset her and affect her eating.
On the other hand, gagging more than a few times a meal may mean you are moving too quickly with textures. Try pureeing foods or mashing them a bit more and moving on with more texture later. Some kids like to take big bites (my daughter) and might not do well with teething biscuits or cheerios while other kids their age do well. Be responsive to your child. (M would bite off big pieces of Zwieback and gag, so we just didn't offer them. She did better with more melt-in-your-mouth foods like buttered toast pieces or graham crackers.)
Tips to reduce gagging and minimize choking risk:
1) Eat at the table, not on the go.
2) Minimize distractions like TV.
3) Offer foods based on what your child can do. (Child Of Mine by Ellyn Satter is a great resource on the details of how to feed, or Born to Eat for more on Baby Led Weaning.)
4) Advance feeding based on your child's development, not age.
5) Have your child sitting upright in a properly supported chair.
6) Avoid high risk choking foods in kids under 3 (nuts, hot dogs, grapes, popcorn), cut grapes in quarters and halves as kids progress.
7) Always be with your child when eating and learn CPR in case she is choking.
Some gagging is not normal.
If your child has a medical history of feeding problems, physical problems, frequent pneumonia or infections that might indicate foods are slipping down the wrong pipe, Downs Syndrome, autism, poor growth, rapid dropping of previously accepted foods, you should look into things further with your child's doctor.
If your older child is gagging over new foods and you are engaged in feeding battles, you need help. This is scary and a very negative experience. Never make a child eat a food he has gagged up onto the plate. There may be some sensory integration issues or other issues that are contributing. (Update, read my book Helping Your Child with Extreme Picky Eating for more!)
If your child has always had problems feeding, something else might be going on. Always talk to your child's health provider if you have concerns, or just aren't sure. You might consider getting video of your child gagging to bring to your doctor if you have a concern. A picture is worth a thousand words.