There are two main competing schemata of categorisation of Celtic languages

Celtic languages

The Celtic languages are descended from Proto-Celtic, or "Common Celtic"; a branch of the greater Indo-European language family...

. The older P-Celtic/Q-Celtic hypothesis links Gaulish with Brythonic as P-Celtic and links Goidelic with Celtiberian

Celtiberian language

Celtiberian is an extinct Indo-European language of the Celtic branch spoken by the Celtiberians in an area of the Iberian Peninsula lyingbetween the headwaters of the Duero, Tajo, Júcar and Turia rivers and the Ebro river...

 as Q-Celtic. The difference between P and Q languages is the treatment of Proto-Celtic

Proto-Celtic language

The Proto-Celtic language, also called Common Celtic, is the reconstructed ancestor language of all the known Celtic languages. Its lexis can be confidently reconstructed on the basis of the comparative method of historical linguistics...

 *k^w, which became *p in the P-Celtic languages but *k in Goidelic. For example the word for head is pen in Brythonic languages but ceann in Goidelic; the word for son is mab (earlier map) in Brythonic but mac in Goidelic – maqq on the Primitive Irish Ogham

Ogham

Ogham is an Early Medieval alphabet used primarily to write the Old Irish language, and occasionally the Brythonic language. Ogham is sometimes called the "Celtic Tree Alphabet", based on a High Medieval Bríatharogam tradition ascribing names of trees to the individual letters.There are roughly...

 inscriptions.

The other scheme, defended for example by McCone (1996), links Goidelic and Brythonic together as an Insular Celtic

Insular Celtic languages

Insular Celtic languages are those Celtic languages that originated in the British Isles, in contrast to the Continental Celtic languages of mainland Europe and Anatolia. All surviving Celtic languages are from the Insular Celtic group; the Continental Celtic languages are extinct...

 branch, as opposed to Continental Celtic languages including Gaulish

Gaulish language

The Gaulish language is an extinct Celtic language that was spoken by the Gauls, a people who inhabited the region known as Gaul from the Iron Age through the Roman period...

 and Celtiberian

Celtiberian language

Celtiberian is an extinct Indo-European language of the Celtic branch spoken by the Celtiberians in an area of the Iberian Peninsula lyingbetween the headwaters of the Duero, Tajo, Júcar and Turia rivers and the Ebro river...

.

P-Celtic incorporates the following:

• Gallic
  • Gaulish
    Gaulish language
    The Gaulish language is an extinct Celtic language that was spoken by the Gauls, a people who inhabited the region known as Gaul from the Iron Age through the Roman period...
    
    
  • Lepontic
    Lepontic language
    Lepontic is an extinct Alpine language that was spoken in parts of Rhaetia and Cisalpine Gaul between 550 and 100 BC. It was a Celtic language, although its exact classification within Celtic has been the object of debate...
    
    
  • Noric
    Noric language
    The Noric language or Eastern Celtic language was a Continental Celtic language. It is attested in only two fragmentary inscriptions from the Roman province of Noricum , which do not provide enough information for any conclusions about the nature of the language to be drawn...
    
    
  • Galatian
    Galatian language
    Galatian is an extinct Celtic language once spoken in Galatia in Asia Minor from the 3rd century BC up to at least the 4th century AD, although ancient sources suggest it was still spoken in the 6th century....
    
    
• Pritennic
  • Pictish
    Pictish language
    Pictish is a term used for the extinct language or languages thought to have been spoken by the Picts, the people of northern and central Scotland in the Early Middle Ages...
    
    
• Brythonic
  Brythonic languages
  The Brythonic or Brittonic languages form one of the two branches of the Insular Celtic language family, the other being Goidelic. The name Brythonic was derived by Welsh Celticist John Rhys from the Welsh word Brython, meaning an indigenous Briton as opposed to an Anglo-Saxon or Gael...
  
   (British)
  • West Brythonic
    • Old Welsh
      Old Welsh language
      Old Welsh is the label attached to the Welsh language from about 800 AD until the early 12th century when it developed into Middle Welsh. The preceding period, from the time Welsh became distinct from the British language around 550, has been called "Primitive Welsh".Many poems and some prose...
      
      
      • Middle Welsh
        Middle Welsh language
        Middle Welsh is the label attached to the Welsh language of the 12th to 14th centuries, of which much more remains than for any earlier period. This form of Welsh developed from Old Welsh....
        
        
        • Welsh
          Welsh language
          Welsh is a member of the Brythonic branch of the Celtic languages spoken natively in Wales, by some along the Welsh border in England, and in Y Wladfa...
          
          
    • Cumbric
      Cumbric language
      Cumbric was a variety of the Celtic British language spoken during the Early Middle Ages in the Hen Ogledd or "Old North", or what is now northern England and southern Lowland Scotland, the area anciently known as Cumbria. It was closely related to Old Welsh and the other Brythonic languages...
      
      
  • Southwestern Brythonic
    Southwestern Brythonic language
    The Southwestern Brythonic languages are the Brythonic Celtic tongues spoken in Southwestern Britain and Brittany since the Early Middle Ages. During the period of their earliest attestation, the languages appear to be indistinguishable, but eventually they evolved into the Cornish and Breton...
    
    
    • Old Breton
      • Middle Breton
        • Breton
          Breton language
          Breton is a Celtic language spoken in Brittany , France. Breton is a Brythonic language, descended from the Celtic British language brought from Great Britain to Armorica by migrating Britons during the Early Middle Ages. Like the other Brythonic languages, Welsh and Cornish, it is classified as...
          
          
    • Old Cornish
      • Middle Cornish
        • Cornish
          Cornish language
          Cornish is a Brythonic Celtic language and a recognised minority language of the United Kingdom. Along with Welsh and Breton, it is directly descended from the ancient British language spoken throughout much of Britain before the English language came to dominate...
          
          

Q-Celtic incorporates the following:

• Goidelic
  Goidelic languages
  The Goidelic languages or Gaelic languages are one of the two branches of the Insular Celtic languages, the other consisting of the Brythonic languages. Goidelic languages historically formed a dialect continuum stretching from the south of Ireland through the Isle of Man to the north of Scotland...
  
  
  • Primitive Irish
    Primitive Irish language
    Primitive Irish or Archaic Irish is the oldest known form of the Goidelic languages. It is known only from fragments, mostly personal names, inscribed on stone in the ogham alphabet in Ireland and western Great Britain from around the 4th century to 7th or 8th century.-Characteristics:Transcribed...
    
    
    • Old Irish
      Old Irish language
      Old Irish is the name given to the oldest form of the Goidelic languages for which extensive written texts are extant. It was used from the 6th to the 10th centuries, by which time it had developed into Middle Irish....
      
      
      • Middle Irish
        Middle Irish language
        Middle Irish is the name given by historical philologists to the Goidelic language spoken in Ireland, Scotland and the Isle of Man from the 10th to 12th centuries; it is therefore a contemporary of late Old English and early Middle English...
        
        
        • Irish
          Irish language
          Irish , also known as Irish Gaelic, is a Goidelic language of the Indo-European language family, originating in Ireland and historically spoken by the Irish people. Irish is now spoken as a first language by a minority of Irish people, as well as being a second language of a larger proportion of...
          
          
        • Scottish Gaelic
          Scottish Gaelic language
          Scottish Gaelic is a Celtic language native to Scotland. A member of the Goidelic branch of the Celtic languages, Scottish Gaelic, like Modern Irish and Manx, developed out of Middle Irish, and thus descends ultimately from Primitive Irish....
          
          
        • Manx
          Manx language
          Manx , also known as Manx Gaelic, and as the Manks language, is a Goidelic language of the Indo-European language family, historically spoken by the Manx people. Only a small minority of the Island's population is fluent in the language, but a larger minority has some knowledge of it...
          
          
• Celtiberian languages
  • Celtiberian
    Celtiberian language
    Celtiberian is an extinct Indo-European language of the Celtic branch spoken by the Celtiberians in an area of the Iberian Peninsula lyingbetween the headwaters of the Duero, Tajo, Júcar and Turia rivers and the Ebro river...
    
    
  • Northwestern Hispano-Celtic
    Gallaecian language
    The Northwestern Hispano-Celtic, Gallaecian or Gallaic, is classified as a Q-Celtic language under the P-Q system and was closely related to Celtiberian...