The most popular and easiest growing under water or submerged oxygenating pond plant is anacharis (Egeria densa) though followed closely by a wonderful hornwort which shall be discussed soon. More info below the big picture.
Contrary to popular belief you probably do not need these plants in your water garden. A single pump of almost any size running even a spitter or tiny waterfall will add probably a million times more oxygen to the pond than the plants in an hour. HOWEVER there are numerous other benefits from this plant and I do suggest you having plenty of these plants at the bottom of your water garden. This plant is one of only about five that should be in your pond and not be planted in soil. A basket with pea gravel should be fine if you have no large koi (14″+) that will shred them.
Anacharis come in small bunches about 6-8 stems rubber banded together. Many on-line companies offer these with optional weights so they sink straight to the bottom and no planting or baskets are required. The weights are a malleable (bendable) heavy metal that you simply wrap around the area where the rubber band is already located at the bottom of the stems. These plants rob the water of excess nutrients rapidly and that is a great thing, they are a super filter and suck up fish waste like it was nothing. The second benefit, a superior hiding place for adult fish and baby fish fry. Baby fry grow up around anacharis and hornwort and stay hidden from larger fish that would eat them. The larger fish hide between anacharis clusters when Hawks, Raccoons, Possums, neighborhood kids with nets, or the big daddy pond nemesis the Heron come around. Having enough submerged grasses and also surface plants (water lettuce, water hyacinths, and water lilies) keep your pond from becoming the daily buffet. Anacharis should be added 1 bunch per 10 gallons of water in ponds under 5000 gallons, or 1 bunch per 20 gallons for ponds over 5000 gallons. Thats many bunches but this product is sold in quantity and is generally a one time investment. If you have enough surface plants in your pond and submerged plants you will never add a drop of chemicals to ward off algae saving you 1000′s of dollars over the years, 100′s per season so just do it right the first time.
Anacharis is winter hardy, sometimes fish nibble on it late or early in the year making it look like theres less in the spring butit quickly rebounds in generally days once thespring water temps are in the 60s. Hornwort can be used as an alternative also winter hardy.