The plant is commonly called a Prayer Plant (Maranta leuconeura), although it has other names. In its native environment, Maranta is a tropical under-story plant, so it likes indirect sunlight, high humidity, relatively high temperatures and no cold drafts.
If you've owned this plant for several months or more, whatever you're doing is working because the second photo appears to show flowers; if you've just purchased or received it as a gift, then the flowers indicate that the plant was well-taken-care of before you got it. Given the right conditions, Marantas are tough plants, with few if any disease or insect problems. The browning on the leaves is not dangerous, just unsightly.
This site contains instructions for caring for the plant (including watering instructions and how to maintain high humidity) as well as a note that leaf burn occurs under high fluoridation or over-fertilization. If you're watering the plant with tap water from a city's water system, you may want to switch to filtered or purchased water to remove the fluoride from the water and see if the leaf burn goes away. If not, then fertilization could be the cause, so don't fertilize for awhile. When you start again, use a timed-release fertilizer like Osmocote.