Sandhi in Sanskrit

Sandhi is when two words join and some letter are modified in the process - a crude definition.

e.g.,
गज​+आनन = गजानन​

So when अ is placed with आ, it becomes आ.

This is just one type of Sandhi - incidentally called Savarna deergha sandhi.(सवर्ण दीर्घ् सन्धि) There are others as well (more difficult ones)

सवर्ण दीर्घ् सन्धि

When  अ इ उ or ऋ are followed by same vowel or long vowel of same type, then in place of these two vowels, one long vowel is substituted. ( savarna - same type 

deergha - long)

Let us look at more examples.

गिरि + इन्द्र  = गिरीन्द्र
वधु+ उक्ति = वधूक्ति
एव + अयम् = अवायम्
कटु + उक्ति = कटूक्ति 

सुख + आसनम् = सुखासनम् 

सिन्धु + उदकम् = सिन्धूदकम् 

Guna sandhi -  गुण सन्धि

When अ /आ vowels are followed by इ/ ई, they are substituted by a single ए vowel,

when  अ /आ vowels are followed by उ /ऊ, they are substituted by a single ओ vowel  and when   अ /आ vowels are followed by ऋ / ॠ, they are substituted by  अर् 

राज + ईश्वरि = राजेश्वरि
महा + इन्द्र = महेन्द्र​
नर + एन्द्र = नरेन्द्र​
महा + उन्नत = महोन्नत
पर​+उपकार = परोपकार​
लम्ब + उदर = लम्बोदर 
च​+ इति = चेति 

महा + ऋषि = महर्षि
सर्वस्य + उपरि = सर्वस्योपरि