Map from Google - Qatar

The climate of Qatar is desert, with a very mild winter and a very hot and sunny summer.
The rainfall occurs during the winter months, and it amounts to less than 100 millimetres per year. Every now and then, some rains may occur in spring, in March, April and even May, while in summer it never rains. The rains in Qatar occur in the form of brief showers, which can sometimes fill the wadis, the river beds which remain dry most of the year, and cause even flooding.
Here is the average precipitation in Doha.
Average precipitation - Doha
DohaJanFebMarAprMayJunJulAugSepOctNovDecYear
DohaJFMAMJJASONDY
Prec. (mm)131716940000131275
Days2221000000019

In the capital Doha the daily average temperature goes from 17.5 degrees Celsius in January, to 35 °C in July. Being Qatar a small and flat country, it has a uniform climate throughout the territory.
Here are the average temperatures of Doha.
Average temperatures - Doha
DohaJanFebMarAprMayJunJulAugSepOctNovDec
DohaJFMAMJJASOND
Min (°C)131417212528292927232015
Max (°C)222327323841424139353024

In winter, from December to February the temperatures are mild and the sky is often clear, so it's a spring-like weather. Lows are around 13/15 °C, and highs around 22/24 °C. However, there can be some cool days, especially in January, in which the temperature drops below 10 °C at night, and stay around 15/16 °C in the afternoon.
In spring the temperature rises fairly quickly, so much so that already in April highs are around 32 °C. Spring is also the season in which the warm wind blows more often from the Arabian desert, bringing dust and sand storms, and sudden increases in temperature, which can exceed 40 °C in April; sometimes this happens even in autumn, especially in October.
Summer from May to September is very hot, with highs from 38 to 42 °C, and lows around 25/29 °C. There are days in which the wind from the inland desert prevails, so that the daytime temperatures rise above 40 °C, and sometimes even 45/47 °C, while other periods are less hot but wetter, with highs around 37/38 °C, which can be even worse because they are sweltering. Over the course of the summer months, the relative humidity gradually increases, so that the second part of the season, in August and September, is particularly stifling. The combination of moisture and temperature that is observed in summer in the countries of the Persian Gulf, is actually one of the worst in the world.
Autumn, usually sunny, is still hot in October, when the maximum temperature is around 35 °C, while in November it drops to 28/29 °C.

Doha

The sea is a bit cool from January to March, particularly in February when it reaches the lowest temperature, about 20 °C, while it's still quite warm in December, when it's about 24 °C. In summer, the waters of the Persian Gulf become very warm, exceeding 30 °C from June to October.
Sea temperature - Doha
DohaJanFebMarAprMayJunJulAugSepOctNovDec
DohaJFMAMJJASOND
Sea (°C)222021232730323333312824

When to go

The best time to visit Qatar runs from mid-November to mid-March: it is the only time of the year which is not hot. In general, the weather is mild and sunny from December to February, warm but not too much in November and March. Some cloudy days are also possible, sometimes with rain, especially from December to February.

What to pack

In winter: light clothes for the day, sweater and jacket for the evening and for cool days, scarf for the wind.
In summer: lightweight clothes, made of natural fabric (cotton or linen), desert turban, sweatshirt and scarf for air conditioned places. For women, it is better to avoid shorts and miniskirts outside of tourist resorts.
For the reef, snorkeling equipment, water shoes or rubber soled shoes.