Welcome!
You've received great replies above. I'll just add that the definition of "gifted" provides at least a partial answer to your question. Definitions of gifted can be found on many websites, including the US Department of Education (USDOE), National Association for Gifted Children (NAGC), The Davidson Database, Hoagies Gifted Education Page, and others.
There are also lists of commonly observed traits of gifted kids (such as early math and/or reading) on these websites and others.
Individuals who have tested in the intellectually gifted range (generally IQ 132+) may have exhibited early math skills and/or advanced vocabulary however not all precocious children are also intellectually gifted.
Advanced vocabulary can be a sign of giftedness and/or reflect the words and phrases which a child commonly hears in an enriched environment.
Over time, if the use of vocabulary seems to develop in a manner which others may describe as stilted, formal, or pedantic and does not help a child connect with others, this may possibly be a sign that the child is on the autism spectrum. And/or would make a good lawyer someday.
If a child is both gifted and has a learning difference/disability, this called twice exceptional (2e) and the child is eligible for special education services such as remediation/accommodation to help them fully access their education. This can help a child connect well with others to best develop his/her gifts. The Wrights Law website is an excellent source of information.
Testing may help reveal your child's profile.
In general, giftedness can occur in one or more realms: athletic, artistic/creative, leadership, intellectual, and/or specific academic strength. In addition to exceptional gifts, children may have one or more learning differences or learning disabilities therefore be termed twice-exceptional (2e). For the most part, gifted forums focus less on athletic gifts because there are other sources of support for developing those gifts, and focus more on developing the gifts and meeting the needs toward the end of the lists earlier in this paragraph.
That being said, misbehavior must be addressed; being gifted does not explain or justify poor social skills. Testing may help determine whether mismatched curriculum may be at the root of the misbehavior (for example, greater academic challenge keeps your child mentally occupied and behaving) or whether your child may also have a learning difference or learning disability (and may benefit from remediation and/or direct training in social skills). As many topics on the forum tend to repeat, there are old threads on these topics, which may be of interest.
