Alphabet

Alphabet is not difficult and many letters are already familiar to an English speaking person.
  Russian alphabet is a Cyrillic alphabet which was created by two Greek brothers, monks, St.Cyril and Methodius, missionaries to the Slavs. They devised the alphabet for the use by Slavic peoples. At present, basic Cyrillic alphabet is used by several languages. Each language also has some letters which are specific to that particular language. Cyrillic alphabet is used by Russians, Byelorussians, Ukrainians, Bulgarians, Serbians est.

Russian alphabet is phonetic, or nearly phonetic. This means that every SOUND in Russian language has a corresponding LETTER in the alphabet. There are just few spelling rules in Russian.
This means that once you learn how to pronounce Russian letters you can read Russian. You might not understand, but we will be able to read.
If you can read, you can understand already a lot. First we use a lot of foreign words and by being able to read it, you will be able to understand what are we talking about.

Russian alphabet is not difficult and many letters are already familiar to an English speaking person.
There are only 16 completely new letters to be learned and 6 letters to learn new pronounciation.

Russian letters that are (almost) the same.
А а - Pronounced like the "a" in the word "father" or "car". It is not the 'flat' "a" sound you sometimes hear in words like "cat" or "flat".
К к - Pronounced like the "k" in "kitten" or "kangaroo". This letter replaces the english "c" sound in words like "cat".
М м - Pronounced like the "m" in man. (Note: Unlike english, the hand-written "м" should always start from the bottom)
O o - When stressed, it is pronounced like the "o" in "bore". When un-stressed it is pronounced more like the letter "a". (See later notes.)
Т т - Pronounced like the "t" in "tap". (Note: The hand-written form for "т" should always start from the top, as it looks quite similar to the letter "м")

Russian letters that look like english letters but sound different.
(These are the most important to learn so you don't get them mixed up.)
В в - Pronounced like the "v" in "vet". (Equivalent to the english letter "v").
Е е - Pronounced like the "ye" in "yes".
Н н - Pronounced like the "n" in "no". (Equivalent to the english letter "n").
Р р - Pronounced like the "r" in "run", but it is rolled. (Equivalent to the english letter "r").С с - Pronounced like the "s" in "see". (Equivalent to the english letter "s"). (It might help to remember that it's used like the "s" sound in the english words "centre" and "cent".)
У у - Pronounced like the "oo" in "boot" or "root".
Х х - Pronounced like the "h" in "hello". However, this is often pronounced more like the "ch" in the Scottish "Loch" or German "Bach", or the spanish "x" in "Mexico".

Russian letters that look unusual, but have familiar sounds
Б б - Pronounced like the "b" in "bat". (Equivalent to the english letter "b").
Г г - Pronounced like the "g" in "go". (Equivalent to the english letter "g").
Д д - Pronounced like the "d" in "dog". (Equivalent to the english letter "d").
З з - Pronounced like the "z" in "zoo". (Equivalent to the english letter "z").
И и - Pronounced like the "i" in "taxi". (Sometimes equivalent to the english letter "i", the short 'ee' sound.). (Note: The hand-written form for "и" looks a little like the english "u").
Л л - Pronounced like the "l" in "love". (Equivalent to the english letter "l").
П п - Pronounced like the "p" in "pot". (Equivalent to the english letter "p").
Ф ф - Pronounced like the "f" in "fat". (Equivalent to the english letter "f").
Э э - Pronounced like the "e" in "fed".


New Russian letters
(The sounds will be familiar, but they don't have their own letter in English).
Ю ю - Pronounced like the "u" in "universe". (Pronounced much like the english word "you").
Я я - Pronounced like the "ya" in "yard".
Ё ё - Pronounced like "yo" in "yonder".
Ж ж - Like "s" in "measure", "pleasure" or "fusion" or like "g" in colour "beige". (As there is no english symbol for this sound, it is usually represented as "zh")
Ц ц - Similar to the "ts" sound in "sits" or "its".
Ч ч - Pronounced like the "ch" in "chips" or "church" .
Ш ш - Pronounced like the "sh" in shut.

Specific Russian Sounds
Щ щ - Pronounced like "sh" but with your tongue on the roof of your mouth. Try putting your tongue in the same position as you would to say "ch" but say "sh" instead. English speakers may find it hard to define the difference between "ш" and "щ".
Ы ы - Pronounced like the "i" in "bit" or "ill". (Said with your tounge slightly back in your mouth.)
Й й - This letter is used to form diphthongs. So "oй" is like the "oy" sound in "boy" or "aй" is like the "igh" in "sigh".

Pronunciation Symbols
(These letters have no sound on their own, but are still considered letters.).
Ъ ъ - The 'Hard Sign' is rarely used. It indicates a slight pause between sylables.
Ь ь - The 'Soft Sign' makes the previous letter 'soft'. Think of the "p" sound in the word "pew". (Try inflecting a very slight "y" sound onto letter before it.)