Whether you are eligible for veterans benefits and VA health care benefits depends on the type of discharge you received when you left the military.
If you received an honorable discharge, which means you exceeded or met the military's standards for performance and personal conduct, you are eligible for all VA benefits and health care.
A general discharge under honorable conditions means that your service was satisfactory, but did not deserve the highest level of discharge for performance and conduct. Many veterans with this type of discharge may have engaged in minor misconduct. While a discharge under honorable conditions may not be what a veteran wants on his or her resume, it qualifies for VA health care, TRICARE's Continued Health Care Benefit Program (military health insurance), VA disability compensation, VA pension, VA home loans, and all other veterans benefits except for educational benefits under the Montgomery or Post-9/11 GI Bill. (For the Montgomery GI Bill program or Post-9/11 GI Bill program, you need an honorable discharge.)
If you were given an Other Than Honorable Discharge (formerly known as an Undesirable Discharge), you will not likely be eligible for any benefits, but the VA will determine on a case-by-case basis whether circumstances are such that you should be entitled to a certain benefit or class of benefits. When you apply for a benefit, such as disability compensation, the VA will give you a "character of service" review that will determine whether you are eligible. This is different from a discharge upgrade.
Not all veterans with these types of discharge are eligible for a character of service determination (CSD). For instance, those convicted of a felony or spying for the enemy would not be offered a CSD. For more information, see Nolo's article on VA character of service determinations.
If you received a Bad Conduct Discharge that was issued by a Special Court Martial, you may still qualify for certain benefits just as if you had an other than honorable discharge. The VA will determine your eligibility through a character of service determination.
If you received a Bad Conduct Discharge that was issued by a General Court, you are not eligible for any benefits.
If you left the military with a Dishonorable Discharge or a Dismissal (for an officer), you are not eligible for any benefits.
If you had two or more periods of service, and one ended with good paper (honorable or under honorable conditions) and the other ended with "bad paper," there are some complicated rules about whether you are eligible for benefits. Generally, if you received good paper for the first period of service, and "bad paper" for the second, you may be eligible for disability compensation and health care for any disabilities that occurred or were aggravated during your "good" period of service. For more information, see Nolo's article on eligibility for VA benefits with good and bad paper.