Map from Google - Svalbard

The climate in Svalbard is subpolar along the west coast, reached by the last branch of the Gulf Stream, and is polar on the east coast, where the temperature remains around freezing (0 °C or 32 °F) even in summer. In winter, polar ice typically reaches the north and east coasts of the islands, while leaving free the western ones.
No coincidence that the settlements (Longyearbyen, Barentsburg, Pyramiden, Ny-Alesund) are located in the western side of the main island, Spitsbergen, where the temperature in the summer months rises above freezing, albeit by a few degrees.
Here are the average temperatures in Longyearbyen.
Average temperatures - Longyearbyen
LongyearbyenJanFebMarAprMayJunJulAugSepOctNovDec
LongyearbyenJFMAMJJASOND
Min (°C)-20-21-20-16-7-132-3-9-14-18
Max (°C)-13-13-13-9-33761-4-8-11
Min (°F)-4-6-4319303736271670
Max (°F)999162737454334251812

Clashes between air masses of different origin give rise to an unstable and windy weather for much of the year, while in summer the meeting of currents can lead to fog, especially offshore.
The islands are property of Norway, but in the past there were also Soviets settlements (Barentsburg, Pyramiden, the latter now abandoned) and there are still Russians inhabitants.
In inland areas there are some elevations, snow-covered all year round, from which glacial tongues descend, eventually flowing to the sea. The highest peak, Newtontoppen, reaches 1,713 metres (5,620 feet). In summer, you can also ski.
The wind is frequent throughout the year, although it calms down a bit from June to August.
The northernmost tip of Svalbard is only one thousand kilometres (600 miles) away from the North Pole. Being well beyond the Arctic Circle, in the Svalbards the sun never sets for a long period, from the 20th of April to the 25th of August, while it never rises from late October to mid-February. The long polar night, however, can be lit by the aurora borealis.
The islands are a cruise destination, to see fjords and glaciers, or polar bears, whales and seals, as well as the millions of migratory birds that pass in summer.

Beware of polar bears!

Precipitation is not abundant, since it amounts to about 275 millimetres (11 inches) per year within the western fjords, but it's frequent and occurs mainly in the form of light snow. In summer, it takes the form of rain or drizzle, at least in the plains. Along the most exposed coasts and in the mountain slopes, precipitation is more abundant, and can reach 1,000 mm (40 in) per year.
Here is the average precipitation in Longyearbyen.
Average precipitation - Longyearbyen
LongyearbyenJanFebMarAprMayJunJulAugSepOctNovDecYear
LongyearbyenJFMAMJJASONDY
Prec. (mm)203030151520253025202025275
Prec. (in)0.81.21.20.60.60.811.210.80.8110.8
Days15141612109121414141414158

Winter is long (going from October to May) and freezing, with frequent snowfalls, usually in the form of light snow. The average temperature of February in Longyearbyen is -17 °C (1 °F). April and May are still freezing months, but with 24 hours of daylight they are very different from the dark winter months.
The coldest records are very low, -40 °C (-40 °F) in Barentsburg and -45 °C (-49 °F) in Longyearbyen, but in fact these values are reached very rarely, also because the climate at high latitudes in recent decades has warmed significantly: so it's more likely for the temperature to drop, in the coldest days of the year, to -25/-30 °C (-13/-22 °F).
Sometimes the temperature can exceed freezing by a few degrees and it can rain even in winter, and this typically happens when a cold wave in Europe is about to occur: in fact there is a heat exchange, so warm air from the south comes this far, and in return the polar air moves towards the south.
The Bear Island (Bjørnøya), more than 200 km (125 mi) south of the main islands, is much milder in winter, as the average temperature of the January-March period is -7 °C (19 °F), while in summer the difference with the main islands is reduced.

Summer, from June to mid-September, is cold: in June and September the temperature is just above freezing, while in July and August, the warmest months, the average is around 4/5 °C (39/41 °F).
In the warmest days, the temperature can reach 18/20 °C (64/68 °F) inside the largest fjords, as happens in Isfjorden, in which Longyearbyen and Barentsburg are located, while in the "maritime" Ny-Alesund it doesn't go above 13/14 °C (55/57 °F).

The amount of sunshine in Svalbard is never good, however, it's decent in April and May, and then gradually decreases, but remains quite acceptable from June to August: of course, the sky is often cloudy, but there is some chance to see the sun for a few hours, given the length of the days, perhaps even at midnight.

Longyearbyen

The sea is very cold throughout the year; as mentioned, the west coast, where there are the main settlements such as Longyearbyen, is generally free of ice.
Here are the average sea temperatures.
Sea temperature - Longyearbyen
LongyearbyenJanFebMarAprMayJunJulAugSepOctNovDec
LongyearbyenJFMAMJJASOND
Sea (°C)222123565432
Sea (°F)363636343637414341393736

When to go

The only two months in which the temperature is permanently above freezing (0 °C or 32 °F), at least in the plains, are July and August, therefore they are the best months. Of course, you must bring warm clothes.
To visit the Svalbards in their winter appearance, April and May are preferable, as the sun is above the horizon, and therefore it is not dark all day.
The time zone is the same of Norway and Central Europe, and even here there is daylight saving time.