Map from Google - Romania

Romania has a continental climate, with cold winters and hot summers.
The Black Sea coast has a slightly milder climate, but not by much, since it often receives air masses coming from inland.
In the long Carpathian range, which crosses the country, the weather becomes colder and wetter with increasing altitude: in winter, snowfalls are frequent, while in summer, thunderstorms occur in the afternoon.

Precipitation is not abundant in inland plains, since it amounts to around 600 millimetres (23.5 in) per year, although it is well distributed over the seasons, with a maximum in May and June, mainly because of afternoon thunderstorms.
Here is the average precipitation in the capital Bucharest.
Average precipitation - Bucharest
BucharestJanFebMarAprMayJunJulAugSepOctNovDecYear
BucharestJFMAMJJASONDY
Prec. (mm)403540457075655040305045595
Prec.(in)1.61.41.61.82.832.621.61.221.823.4
Days66676676556672

Along the coast of the Black Sea, precipitation is quite low, amounting to around 385 mm (15 in) per annum, being the summer thunderstorms less frequent; here, from 20 to 40 mm (0.8 to 1.6 in) per month fall throughout the year.
Here is the average precipitation in Constanta.
Average precipitation - Constanta
ConstantaJanFebMarAprMayJunJulAugSepOctNovDecYear
ConstantaJFMAMJJASONDY
Prec. (mm)302520303540353025404035385
Prec.(in)1.210.81.21.41.61.41.211.61.61.415.2
Days44344333235543

As mentioned, the situation is different in mountainous areas, where rainfall is more abundant, and exceeds 40 in per year.

Winter, from December to February, is cold throughout Romania. Cold air stagnates in the lowest layers of the atmosphere, and the sky is often cloudy. Snowfall is quite common, but not abundant; in inland plains, there are about thirty days with snow per year. In Bucharest, the ground is often covered with snow. During winter, temperatures are fairly uniform across the continental region, and decrease mainly with increasing altitude. In the plains, the average temperature in January hovers between -1 °C (30 °F) and -2.5 °C (27.5 °F). In Bucharest, the average is -2 °C (28 °F).
Here are the average temperatures in Bucharest.
Average temperatures - Bucharest
BucharestJanFebMarAprMayJunJulAugSepOctNovDec
BucharestJFMAMJJASOND
Min (°C)-6-306111416151162-3
Max (°C)241118232729292518104
Min (°F)212732435257615952433627
Max (°F)363952647381848477645039

In the Transylvanian Plateau, west of the Carpathians, winter is slightly colder: in Cluj-Napoca, 400 metres (1,300 feet) above sea level, the average in January is -3.5 °C (25.5 °F), while in Sibiu, at the same altitude, it's -4 °C (25 °F).
Romania, like other parts of the Balkan Peninsula, is exposed to outbreaks of cold air masses from northern Europe or Russia, most likely in winter and spring. They were more frequent in the past decades, when the climate was colder, however, they can still lower the temperatures to around -20 °C (-4 °F), while in summer, a bit of cool air at high altitudes, in contrast with the hot air lying in the lower layers, can give rise to thunderstorms.
During winter, the mildest plain area of Romania is Dobrudja, the stretch of plain east of the Danube River, which is affected by the influence of the sea, where the average temperature in January is just above freezing, around 1 °C (34 °F). There are on average 110 days per year with night-time temperature below freezing in Bucharest, and 70 in Constanta, on the coast. The cold record of the last 20 years is -23 °C (-9 °F) in Bucharest, and -18 °C (0 °F) in Constanta. On colder winters, the Black Sea along the coast can even freeze, as happened in February 2012. This fact tells us that winter along the coast of the Black Sea, albeit milder than in the interior, is still cold.
Here are the average temperatures in Constanta.
Average temperatures - Constanta
ConstantaJanFebMarAprMayJunJulAugSepOctNovDec
ConstantaJFMAMJJASOND
Min (°C)-3-1271216181814950
Max (°C)45814202426262217116
Min (°F)273036455461646457484132
Max (°F)394146576875797972635243

As mentioned, until a few decades ago, winter was a few degrees colder, here as in other Eastern European countries, so if you look at historical archives and long-term data, you will find average temperatures lower than those of the last decades.

Summer, from June to August, is quite warm in the plains and hills of Transylvania (the region west and north of the Carpathian Mountains) and in the region east of the Carpathians, on the border with Moldova: highs are around 25 °C (77 °F) in the plains, and 23 °C (73 °F) around 400 metres (1,300 feet) above sea level. Wallachia, the region south of the Carpathians where Bucharest is located, is warmer, and it's often hot: here it's easy for the temperature to reach 35 °C (95 °F) and higher. Sometimes, however, it can get hot even in the northern plains and hills.
Along coast of the Black Sea, in summer the weather is nice and pleasantly warm, sometimes even hot, but tempered by the breezes.
Summer is a fairly rainy season, especially because of afternoon thunderstorms, which occur especially in May and July, and are most frequent and abundant in the mountains. Sometimes, these thunderstorms are so strong that they can cause floods.
The amount of sunshine in summer is good in the lowlands, while cloud cover is more frequent in the mountains and in Transylvania, where, however, the sun often shines, especially in the morning.

Bucharest

At the Bran Castle, which according to legend would be Dracula's castle, located at 750 metres (2,450 ft) above sea level, winters are cold and snowy, while summers are pleasant, with highs around 23 °C (73 °F) and cool nights, but also possible thunderstorms in the afternoon. A similar climate is found in the Poenari Castle, located at 820 metres (2,700 feet) of altitude, which should be the real Dracula's Castle, Prince Vlad III of Wallachia.
The Monasteries of Bucovina are located about 350 metres (1,150 ft) above sea level, in Romanian Moldavia, near the border with the Republic of Moldova: during the day it can get hot in July and August, while nights are still cool.

The temperature of the Black Sea is acceptable for swimming in summer: it's around 23 °C (73 °F) in July, and 24 °C (75 °F) in August, while it is a bit cool in June and September.
Sea temperature - Constanta
ConstantaJanFebMarAprMayJunJulAugSepOctNovDec
ConstantaJFMAMJJASOND
Sea (°C)76710152023242117139
Sea (°F)454345505968737570635548

When to go

The best time to visit Romania is from May to September.
June is the rainiest month, because of afternoon thunderstorms.
July and August can be hot in inland plains, especially in the south, ie in Wallachia (see Bucharest), so you can choose May, June and September, especially if you don't like the heat.
In May, the country is in bloom and the temperature is generally pleasant, even though afternoon thunderstorms are already possible.
September is usually a sweet and quiet month, with little rain; sometimes, a warm and sunny weather lasts until October.
The Black Sea coast in summer is good for a beach holiday, but in winter, having an average temperature around freezing and possible wind and snowfalls, it's certainly not a refuge from the cold European winter.

What to pack

In winter: warm clothes, sweater or fleece, down jacket, hat, gloves and scarf, raincoat or umbrella.
In summer: light clothes, t-shirts and shorts, but also long pants, light jacket and sweatshirt for the evening; umbrella. Some heavier clothes for the mountains.