From the Guidelines

Calcium carbonate is the best calcium supplement due to its high bioavailability and cost-effectiveness.

Bioavailability: Both calcium citrate and calcium carbonate are highly bioavailable sources of calcium, but calcium carbonate is cheaper 1.
Absorption: Calcium carbonate may be less readily absorbed when taken between meals, while calcium citrate bioavailability is not affected by meals 1.
Cost: Calcium carbonate is likely to be the most cost-effective choice in most settings 1.
Specific Populations: However, for certain populations such as bariatric patients, calcium citrate may be preferable due to its absorption being independent of stomach acidity 1.
Dosing: To aid calcium absorption, it is recommended to take calcium in equally divided doses, with calcium carbonate taken with food and calcium citrate taken with or without food 1.
Supplement Formulation: Calcium citrate may be the preferred supplement for people at risk of developing kidney stones 1.

The recommended daily intake of calcium is at least 1000 mg per day for individuals under 50 years of age and at least 1200 mg per day for those older than 50 years 1.

From the Research

Calcium citrate and calcium carbonate are the most common forms of calcium supplements 2
Calcium citrate can be taken without food and is the supplement of choice for individuals with achlorhydria or who are taking histamine-2 blockers or protein-pump inhibitors 2
Calcium lactate and calcium gluconate are less concentrated forms of calcium and are not practical oral supplements 2

The combination of calcium citrate and vitamin D is effective in reducing the incidence of non-vertebral and hip fractures 3
Calcium supplementation has been linked to both increased and decreased cardiovascular disease risk, creating considerable uncertainty 4
The evidence for calcium supplementation effects to both reduce risk of fracture and increase coronary heart disease and mortality are reviewed, with recent reports shedding uncertainty over the effectiveness of calcium supplements to reduce risk of fracture 4

The maximum dose of elemental calcium that should be taken at a time is 500 mg 2
The current Recommended Dietary Allowances for calcium still appear to be a good target, with potential risks for chronic disease if intakes fall too short or greatly exceed these recommendations 4
Calcium supplements have only a minor place in contemporary medical practice, with the United States Preventive Services Task Force not supporting the use of calcium supplements in healthy community-dwelling adults 5

Potential adverse effects of calcium supplementation include gastrointestinal complaints 2
Renal calculi in most studies have not been associated with calcium supplementation 2
The risk of advanced and fatal prostate cancer has been associated with calcium intakes from food or supplements in amounts >1500 mg/d 2
Recent data suggested potential adverse cardiovascular effects from calcium supplementation 6