Assuming you are eligible for premium tax credits, the amount of your credit will be calculated based on how you file your taxes. If, for example, you claim your partner and your children as tax dependents, you will be considered a household of four when you apply for subsidies. As another example, if you and your partner file taxes separately and you each claim one of your children, you each will be considered as a household of two. The income of each household would be evaluated separately to calculate eligibility for and the amount of premium tax credits and cost-sharing reductions. Consult your tax adviser for more information about how to file your taxes.
As for the type of coverage your family can purchase, that may depend on where you live. Some states recognize and register domestic partnerships. In other states, domestic partnerships can be recognized and registered only in certain cities or counties. Generally, insurers will offer family policies to married couples, but some may offer family policies to domestic partners who are not married. If you buy one policy for the entire family, all the tax credits you are eligible for can be used to reduce the premium for that policy. If you buy separate policies, you can allocate the premium tax credits across two plans.