Hi there heather0104,
If you're using iCloud to sync your contacts, make sure that the option is enabled under Settings > [your name] > iCloud. Likewise, if you sync with other services (like your email provider, or a work/school account) go into Settings > Contacts > Accounts and make sure the correct account is active. If the feature to sync contacts is turned off, the contact names may not appear where expected.
If the issue is particular to certain contacts, make sure you have no inadvertently deleted those individuals. If your'e using iCloud you can restore deleted contacts following Restore contacts stored in iCloud on iCloud.com - Apple Support.
For more help, please get in touch with Apple Support directly, as they will have access to additional tools and resources to further help. They can be reached here: Get Support.
Have a wonderful day.