Dakuten: two strokes that voice a consonant
Two small strokes in the top right corner, called dakuten or ten-ten, change the consonant: k becomes g (か → が), s becomes z (さ → ざ), t becomes d (た → だ), and h, less obviously, becomes b (は → ば). The vowel stays the same, and katakana works the same way (カ → ガ). Watch the s and t rows: じ and ぢ are both read jee, and ず and づ are both read zoo, so ぢ and づ only turn up in a handful of words such as はなぢ (nosebleed).
Handakuten: the small circle that makes a p
A small circle in the top right corner, called handakuten or maru, goes only on the h row and turns h into p: は → ぱ, ひ → ぴ, ふ → ぷ, へ → ぺ, ほ → ぽ. Katakana does the same, so ホ becomes ポ, as in ポケット (pocket). Japanese p is a little softer than the English one, without the puff of air you put into 'pin'.
Small ゃ, ゅ, ょ: two kana, one syllable
When や, ゆ or よ is written small after a kana ending in i, the two merge into a single syllable and the i disappears. き plus ゃ gives きゃ, kyah, one beat rather than kee-yah. After し, ち and じ even the y is swallowed, so しゃ is shah, ちょ is choh and じゅ is joo. Size matters: きや, with a full-size や, really is two syllables. Katakana follows the same pattern, as in キャベツ (cabbage).
Small っ: a doubled consonant
A small っ (ッ in katakana) has no sound of its own; it tells you to hold the next consonant for an extra beat before releasing it. Compare きて (come) with きって (stamp): the second has a short silent catch, like the k in the middle of 'bookcase'. Romaji shows it by doubling the consonant, which is why you see kitte, zasshi and Hokkaido. Getting the length wrong can change the word, so give that pause its full beat.
Long vowels: ー in katakana, an extra vowel in hiragana
Japanese counts vowel length, so a held vowel takes two beats and can change the meaning: おばさん is aunt, おばあさん is grandmother. Katakana marks length with a dash, ー, as in コーヒー (coffee), where both vowels are held. Hiragana writes the extra vowel out instead, as in おかあさん (mother) and おにいさん (big brother). Long o is usually spelled with う rather than お, so とうきょう (Tokyo) has two long o's and no oo sound at all, and long e is often spelled with い, as in せんせい (teacher).
ん: an n that adapts to what follows
ん (ン in katakana) is a syllable on its own and gets a full beat, so せんせい has four beats: seh-n-seh-ee. Before b, p and m it closes the lips into an m, as in さんぽ (a walk) and しんぶん (newspaper). Before k and g it becomes the ng of 'sing', as in りんご (apple). At the end of a word it's a soft nasal hum made at the back of the mouth, with the tongue tip resting low instead of touching the teeth. Before a vowel or y, keep it as a beat of its own, so that きんえん (no smoking) doesn't blur into きねん (commemoration).
Extended katakana: small vowels for foreign sounds
To write sounds that native Japanese words don't have, katakana pairs a regular character with a small ァ, ィ, ゥ, ェ or ォ, and the two are read as one syllable. That gives f before every vowel (ファ, フィ, フェ, フォ), a t and d before ee without the usual ch and j sound (パーティー, party), and sheh, jeh and cheh (シェフ, chef). ヴ, which is ウ with the voicing marks, stands for v, though most speakers say it as b, so ヴァイオリン and バイオリン are the same violin.
Particles: は, へ and を
Three kana are read differently when they work as grammar particles. は, normally hah, is read wah when it marks the topic, so わたしは (as for me) ends in wah. へ, normally heh, is read eh when it means 'to' or 'towards', as in にほんへ (to Japan). を sounds exactly like お and is used for nothing but marking the object of a verb, as in みずをのむ (to drink water). Inside ordinary words は and へ keep their usual sound, as in はな (flower) and へや (room).