How does it work if I claim one of the dependents, have two children, in a divorce situation and my ex tries to claim as well

The IRS looks at who the custodial parent is and who is not.  The custodial parent is who the child lives with for more time during the tax year.​​​​​​​ 

When you allow a non-custodial parent to claim a child, you are only releasing the ability to claim the child for Child Tax Credit under the special rule for divorced and separated parents. 

The custodial parent retains tax benefits related to Head of Household filing status, Earned Income Credit, and the Child and Dependent Care Credit.  To be eligible for these credits, the child must have lived with you for more than half the year, so the non-custodial parent is not eligible for those credits. 

This article will give you more information: Divorced and Separated Parents  

If two people claim the same dependent the IRS will use the TieBreaker Rules 

Sometimes a child meets the rules to be a qualifying child of more than one person. The following rules must be applied to determine who can claim the child as a qualifying child. Under the tie-breaker rule, the child is treated as a qualifying child: 

• The parent, if only one of the persons is the child's parent, 
• The parent with whom the child lived the longest during the tax year, if two of the persons are the child's parent and they do not file a joint return together, 
• The parent with the highest AGI if the child lived with each parent for the same amount of time during the tax year, and they do not file a joint return together, 
• A non-parent, if no parent claims the child as a qualifying child although he or she may do so and only if the non-parent's AGI is higher than the highest AGI of any parent who may claim the child, or 
• The person with the highest AGI, if none of the persons is the child's parent. 

This information is found in Publication 501, Dependents, Standard Deduction, and Filing Information.