Has your blood pressure been too high — meaning you have a systolic pressure of 140 or higher if you're under 60 and of 150 or higher if you're over 60, and a diastolic pressure of 90 or over — for a while? Your doctor may first recommend certain lifestyle changes, like cutting way down on salt, being more physically active, and quitting smoking.  Should those steps not lower your blood pressure to safe levels, however, your physician's next step will be to prescribe you some antihypertensives (blood pressure drugs).
It happened to me about a decade ago, and while I was only very vaguely familiar with the kinds of blood pressure meds out there because my parents were on them too, the fact that not all antihypertensives work quite right for everyone means I've since been acquainted with nearly the whole "who's who" of blood pressure drugs, all in a process of trial and error. You might happen to be prescribed the right antihypertensive(s) for you right away, or you may go through a similar process. If your medication regime is like a passing parade, it's especially hard to keep track on what you should and shouldn't do while taking a particular drug.
Antihypertensives: There's An Awful Lot Out There
The different classes of hypertension drugs all work to reduce your blood pressure in different ways. We won't get into all that here, beyond noting that those patients who are new to a particular class of antihypertensives should ask their prescribing doctors to explain how they work, if they're interested — and should always ask about potential side effects, drug interactions, and any "do's and don'ts" that might come with the medication.
You may be prescribed one blood pressure medication, or a combination of several if your blood pressure is extremely high or if you're found not to react adequately to a single drug.
The classes of antihypertensives are (brace yourselves!):
- Beta bockers
- Angiotensin-converting enzyme inhibitors (ACE inhibitors)
- Angiotensin II receptor blockers (ARBs)
- Calcium channel blockers
- Alpha blockers
- Renin inhibitors
- Centrally acting drugs 
Out of these, the first five are prescribed most commonly. Within each class, there's a whole bunch of generic and brand-name drugs. Each will have its very own side effects, which vary from person to person.
Whatever you're prescribed, it's important to either read the physical package insert that will come with your hypertension med(s), or look the same information up on the internet. If you also take another prescription medication, your doctor will check for possible interactions and make sure not to prescribe meds that could "clash" in some way. The ultimate responsibility for making sure that you do not take over-the-counter drugs, including painkillers, that could interact with your antihypertensive drug lies in your own hands, however.
Taking Hypertension Drugs? Stay Away From Ibuprofen And Other NSAIDs!
It's well-known that ibuprofen, a very popular over-the-counter analgesic and anti-inflammatory drug, reduces the efficacy of rather a few different classes of antihypertensives, namely beta-adrenergic blockers, alpha-adrenergic blockers, diuretics and angiotensin-converting enzyme inhibitors. According to one study: "It has been estimated that the avoidance of minor changes in systolic pressure in patients with osteoarthritis subjected to treatment with nonsteroidal anti-inflammatory drugs would avoid over 30,000 deaths due to myocardial infarction, and over 2000 deaths due to coronary disease, in the United States alone." 
In addition, ibuprofen also generally increases a person's risk of heart attack and stroke, especially upon prolonged use, whether or not the person has hypertension.  This makes it a poor choice for folks with high blood pressure, and although my down-to-earth family doctor told me that I "probably won't die" if I use ibuprofen together with ACE inhibitors and diuretics once a month, I'd personally rather not take the risk — and I wouldn't advise you to, either.
Some other non-steroidal anti-inflammatory drugs are even worse than ibuprofen, though. Definitely don't take indomethacin, naproxen, piroxicam or rofecoxib while you are using blood pressure drugs, especially if you're on an ACE inhibitor and/or a diuretic. 
What Painkillers Can You Use While On Blood Pressure Drugs?
Generally speaking, Aspirin, well-known as a blood thinner but also a member of the NSAID family, has been found to be fairly safe if you take it very rarely. Suggestions that Aspirin also, itself, acts as an antihypertensive have not been found to hold much water however, so never use even low-dose Aspirin on a daily basis without your doctor's explicit instructions. 
Acetaminophen, which is very often discussed as a safer alternative to ibuprofen and Aspirin, can also raise your blood pressure , to the point where the authors of one study point out that "we [may] have a misplaced confidence in the cardiovascular safety of paracetamol". Further study, they conclude, is clearly needed.